Python 3, 133 bytes

Uses a dictionary to make up for a lack of switch-case syntax in Python. See more here.

a="0"*8
for i in input():a={"!":''.join(str(1-int(b))for b in a),"<":a[1:]+"0",">":"0"+a[:-1],"@":a[4:]+a[:4]}.get(i,a)
print(int(a,2))

The accumulator is a string which is converted into a base 10 number at the end.

Example I/O:

$ python3 bitnbytes.py
!
255
$ python3 bitnbytes.py
!>>
63
$ python3 bitnbytes.py
!<@
239
$ python3 bitnbytes.py
!nop!&6*!
255

Pyth, 36 35 bytes

u%@[t_G/G2yGi_jGJ16JG)x"!><@"H256z0

Test harness

The internal representation of the accumulator is an integer. This integer is mod-ed by 256 on each iteration, as desired. The operations performed are -G-1, G/2, G*2 and G converted to base 16, reversed, and converted back to base 10, where G is the accumulator.

I missed the line about ignoring everything else. This has been remedied. Thanks, @Dennis.

Javascript (ES6), 80 91 90 bytes

a=>[...a].reduce((p,c)=>c=='!'?p^255:c=='<'?p*2%256:c=='>'?p>>1:c=='@'?p/16|0+p%16*16:p,0)

Pretty much as short as it can get. Defines an anonymous function which takes the program as input.

• For !, takes x XOR 255, as JS's ~ would consider x a 32-bit number.
• For <, multiplies x by 2 and takes the result mod 256.
• For >, truly shifts the bits of x 1 bit to the right.
• For @, floors x/16 and adds it to x%16*16.

Thanks to @vihan for suggesting the use of reduce to save a byte.

Rust, 121 115 bytes

fn r(s:&str)->u8{let mut n=0u8;for t in s.chars(){match t{'!'=>n=!n,'>'=>n/=2,'<'=>n<<=1,'@'=>n=n>>4|n<<4,_=>()}}n}

Sample run:

fn main() {
    println!("{}", r("!"));    //=> 255
    println!("{}", r("!>>"));  //=> 63
    println!("{}", r("!<@"));  //=> 239
}

Ungolfed:

fn run_ungolfed(s: &str) -> u8 {
    let mut n = 0u8;
    for t in s.chars() {
        match t {
            '!' => n = !n,
            '>' => n >>= 1,
            '<' => n <<= 1,
            '@' => n = (n >> 4) | (n & 15) << 4,
            _ => ()
        }
    }
    n
}

Surprisingly short for Rust. Nothing else really interesting other than the fact that I learned more precedence rules today—who knew (a>>b)|c is the same as a>>b|c?

Shaved off a byte by changing n>>=1 to n/=2; however, the same cannot be done with multiplication, because arithmetic overflow is a panic (i.e. crash) in Rust.

CJam, 37 bytes

0q{"!><@"#"~ 2/ 2* GmdG*+ "S/=~255&}/

Try it online in the CJam interpreter.

How it works

0                   e# Push 0 (accumulator).
q                   e# Read from STDIN.
{                   e# For each character in the input:
  "!><@"#           e#   Find its index in "!><@" (-1 if not found).
  "~ 2/ 2* GmdG*+ " e#   Push that string.
  S/                e#   Split at spaces to push ["~" "2/" "2*" "GmdG*+" ""].
                    e#     "~"      : signed 64-bit bitwise NOT
                    e#     "2/"     : divide by 2
                    e#     "2*"     : multiply by 2
                    e#     "GmdG*+" : (x) -> (x/16) (x%16) -> (16(x%16) + (x/16))
                    e#     ""       : NOOP
  =~                e#  Select the corresponding string and evaluate it.
  255&              e#  Zero all but the 8 least significant bits.
}/                  e#

Python 3, 127 bytes

Edit: shorting, thanks @Jakube

Edit2: fix, thanks @Anachor

a=0
for i in input():a=(a^255if i=="!"else a>>1if i==">"else a<<1if i=="<"else(a&15)<<4|(a&240)>>4if i=="@"else a)&255
print(a)

Python 3, 124 94 93 bytes

a=0
for i in input():
 if i in"!><@":a=(i=='!')*(255-a)+(i==">")*a//2+(i=="<")*(a+a)%256+(i=="@")*(16*(a%16)+a//16)
print(a)

"!" is same as subtracting from 255.
"<" is same as multiplying by 2. But 8 bit register means mod 256.
">" is same as integer division by 2.
"@" means shifting last 4 bits (a%16) by 4 bits(*16) and adding the first four bits(a/16).

EDIT (read shameless copying)
Saw the other answer in python (by Beta decay). It uses a really effective way to simulate switch cases using dictionary. Using that we can write

a=0
for i in input():a={"!":255-a,"<":a<<1&255,">":a//2,"@":(a%16)<<4+a>>4}.get(i,a)
print(a)

Thanks, Beta Decay.

STATA, 197 bytes

di _r(a)
gl b=0
forv x=1/`=length("$a")'{
gl c=substr("$a",`x',1)
if"$c"=="!" gl b=255-$b
if"$c"==">" gl b=int($b/2)
if"$c"=="<" gl b=mod($b*2,256)
if"$c"=="@" gl b=mod($b,16)*16+int($b/16)
}
di $b

Ungolfed

display _request(a) //get the input via prompt and put in var a
global b=0 //initialise A to be 0
forv x=1/`=length("$a")'{ //for loop from 1 to last char in a
global c=substr("$a",`x',1) //get the char at index x in a
if "$c"=="!" global b=255-$b //invert is the same as 255-A
if "$c"==">" global b=int($b/2) //right shift is the same as A/2 (with integer division)
if "$c"=="<" global b=mod($b*2,256) //left shift is the same as A*2%256
if "$c"=="@" global b=mod($b,16)*16+int($b/16) //nibble swap is the same as A%16*16+A/16
}
display $b //display the result of A

Does not work with the online interpreter and requires the non-free default interpreter. This would be somewhat easier with actual bitwise operations, but I don't think they're too useful for most of the common uses of STATA.

Haskell, 89 bytes

a#'!'=255-a
a#'>'=div a 2
a#'<'=mod(a*2)256
a#'@'=mod(a*16)256+div a 16
a#_=a
f=foldl(#)0

Usage example: f "!>>" -> 63

Julia, 117 bytes

p=readline();a=0x0;for c=p c=='!'?(a=~a):c=='<'?(a=a<<1):c=='>'?(a=a>>1):c=='@'?(a=16(a%16)+a÷16):0end;print(int(a))

Ungolfed:

# Read the program as a string from STDIN
p = readline();

# Initialize the accumulator to 0 as an 8-bit unsigned integer
a = 0x0

# Loop over the characters in the input
for c in p
    if c == '!'
        a = ~a
    elseif c == '<'
        a = a << 1
    elseif c == '>'
        a = a >> 1
    elseif c == '@'
        a = 16(a%16) + a÷16
    else
        0 # no-op
    end
end

# Print the value of the accumulator to STDOUT
print(int(a))

Rust, 111 bytes

More of a comment on @Doorknob's answer, but I don't have any rep for comments as I just created an account.

One can shave 10 bytes off his Rust solution with the following:

fn r(s:&str)->u8{let mut n=0u8;for t in s.chars(){n=match t{'!'=>!n,'>'=>n>>1,'<'=>n<<1,'@'=>n>>4|n<<4,_=>n}}n}

C, 107

a;main(c){while(c=getchar()+1)a=c^34?c^61?c^63?c^65?a:(a&240)/16|(a&15)*16:a/2:a*2:a^255;printf("%hhu",a);}

Tidier:

a;
main(c) {
  while(c=getchar()+1)
    a=c^34?
      c^61?
        c^63?
          c^65?
            a
          :(a&240)/16|(a&15)*16
        :a/2:a*2
      :a^255;
    printf("%hhu",a);
}

Basically it's just a collection of nested ternary expressions. I'm incrementing the value obtained from getchar() so that an EOF (-1) results in a value of zero and the program exits.

(ideone link)

Ceylon, 297 290

shared void y(){value t=process.readLine()else"";variable Byte a=0.byte;for(c in t){switch(c)case('!'){a=a.not;}case('>'){a=a.rightLogicalShift(1);}case('<'){a=a.leftLogicalShift(1);}case('@'){a=a.and(#f0.byte).rightLogicalShift(4).xor(a.and(#f.byte).leftLogicalShift(4));}else{}}print(a);}

Formatted:

shared void y() {
    value t = process.readLine() else "";
    variable Byte a = 0.byte;
    for (c in t) { switch (c)
        case ('!') { a = a.not; }
        case ('>') { a = a.rightLogicalShift(1); }
        case ('<') { a = a.leftLogicalShift(1); }
        case ('@') { a = a.and(#f0.byte).rightLogicalShift(4).xor(a.and(#f.byte).leftLogicalShift(4)); }
        else {} }
    print(a);
}

#f and #f0 are hexadecimal numbers for the nibbles, .byte converts an integer into a byte. I'm lucky that Byte's .string attribute already uses the unsigned representation of a byte. Ceylon also features a switch statement without fall-through, and a string is a list of characters, which can be iterated.

I also tried to cut those long shift method names down by using an aliasing import, but this actually becomes 7 bytes longer:

import ceylon.language{Byte{r=rightLogicalShift,l=leftLogicalShift}}shared void x(){value t=process.readLine()else"";variable Byte a=0.byte;for(c in t){switch(c)case('!'){a=a.not;}case('>'){a=a.r(1);}case('<'){a=a.l(1);}case('@'){a=a.and(#f0.byte).r(4).xor(a.and(#f.byte).l(4));}else{}}print(a);}

Formatted:

import ceylon.language {
    Byte {
        r=rightLogicalShift,
        l=leftLogicalShift
    }
}
shared void x() {
    value t = process.readLine() else "";
    variable Byte a = 0.byte;
    for (c in t) {
        switch (c)
        case ('!') { a = a.not; }
        case ('>') { a = a.r(1); }
        case ('<') { a = a.l(1); }
        case ('@') { a = a.and(#f0.byte).r(4).xor(a.and(#f.byte).l(4)); }
        else {}
    }
    print(a);
}

This might be useful if we need those methods a bit more often.

Python 2, 79 bytes

I realized that I have done something very similar to this in Python earlier. This is just a port of my Ruby answer, but it is incidentally the shortest Python answer as of now :D

a=0
for i in raw_input():a=[~a,a/2,a*2,a*16+a/16,a]["!><@".find(i)]&255
print a

Ruby, 81 73 bytes

So much simpler -- no eval! For each valid character in the input, it evaluates each instruction, and finds the appropriate instruction through the index of $& (the current character in the input).

a=0
gets.scan(/[!><@]/){a=[~a,a/2,a*2,a*16+a/16]["!><@".index$&]&255}
p a

Crystal, 139 bytes

def f x
b=0_u8
x.chars.each do|c|
b=case c
when'!'
~b
when'>'
b>>1
when'<'
b<<1
when'@'
b<<4|b>>4
else raise ""
end
end
puts b
end

C# 193

void Main(){byte a=0;foreach(var c in Console.ReadLine()){if(c=='!')a=(byte)~a;if(c=='>')a=(byte)(a>>1);if(c=='<')a=(byte)(a<<1);if(c=='@')a=(byte)(((a&240)>>4)|((a&15)<<4));}Console.Write(a);}

Java, 514 483 411 366 characters

public class T{static int a;public static void main(String[]g)throws Exception{int i;while((i=System.in.read())!='\n'){switch(i){case'!':a=Integer.parseInt(String.format("%8s",Integer.toString(a,2)).replaceAll("[1 ]","0").replaceAll("0","_").replaceAll("_","1"),2);break;case'>':a>>=1;break;case'<':a<<=1;break;case'@':a (a&240)>>4|(a&15)<<4;}}System.out.print(a);}}

Golfed 31 (!) bytes by optimizing the nibble swap with bitwise operations as opposed to lengthy Integer.??? methods.

Golfed 72 (!!!!) bytes by removing the unnecessary string created to swap nibbles. Much better than before!?

Golfed 45 (!!) bytes by removing use of java.util.Scanner and reading from System.in directly. Note that now that the lambda expression is gone, Java 8 is no longer required! Just merely Java 1 would do!

Lua, 344 char

a=string.rep("0",8)
t=io.read()
f={["!"]=function()local s="";for j=1,8 do s=s..(a:sub(j,j)=="0"and"1"or"0") end;return s end,[">"]=function() return "0"..a:sub(1,7) end,["<"]=function()return a:sub(2,8).."0"end,["@"]=function()return a:sub(5,8)..a:sub(1,4)end}
for i=1,#t do a=(f[t:sub(i,i)]or function()return a end)()end
print(tonumber(a,2))

Inspired by @Beta Decay's use of a string accumulator, seeing as lua has no byte type. Could probably be golfed more by using fewer functions.

JavaScript, 104

[].reduce.call(prompt(),function(a,i){return(i=='!'?~a:i=='>'?a/2:i=='<'?a*2:i=='@'?a>>4|a<<4:a)&255},0)

Nested ternary operators map to instructions.

BITWISE AND is used to constrain our Number type to a single byte.