Returning an array of integers with 1000 elements

Your approach is not incorrect. It will get the job done (albeit, slower). If you want to improve your approach, keep reading. But first a few points on your code:

Points on Your code

1. intArrayHolder seems a bit wordy to me. I believe intArray or even array would serve your purpose, without sacrificing readibility. Same for randomNextInt.
2. for (int i = 0; i < intArrayHolder.length; i++) is redundant because the length of intArrayHolder will be equal to size owing to the fact that you declared it like so. for (int i = 0; i < size; i++) will serve you better.
3. Please note that random.nextInt(size + 1) does not return an integer between 1 and 1000. It returns an integer in the range \$[0, 1001)\$, i.e. a number \$x\$ such that \$0\leq x < 1001\$. If you do a count, you will find that you are actually getting \$1001\$ elements. The extra number is \$0\$. To fix it, you must write int randomNextInt = random.nextInt(size) + 1; It is quite easy to understand: you just add a \$1\$ to shift the range from \$[0,1000)\$ to \$[1,1001)\$, i.e. the correct range.
4. ((!isDuplicate) && (randomNextInt != 0)) can be shortened to !(isDuplicate || ramdomNextInt == 0), by using De Morgan's laws. It would be even better if you flip the condition and force next iteration before adding the element to the array.
5. A general advice is to update the loop counter only once. People generally estimate the number of times the loop will execute by stealing a glance at the for (... statement. A while loop is preferred when the number of iterations is unknown or updation is irregular.
6. You seem to be using snake_case along with camelCase. The standard in Java is to use camelCase.

Therefore, the refactored code would be:

import java.util.Random;

public class Domain {

    public int[] randomIntArray(int size) {

        Random random = new Random();
        int array[] = new int[size];

        int i = 0;

        while (i < size) {
            boolean isDuplicate = false;
            int randomInteger = random.nextInt(size) + 1;

            for (int j = 0; j < i; j++) {
                if (array[j] == randomInteger) {
                    isDuplicate = true;
                    break;
                }

            }
            if (isDuplicate) {
                continue;
            } else {
                array[i++] = randomInteger;
            }

        }
        return array;
    }
}

A Better Solution

Your solution's best case running time is \$\Omega(n^2)\$. Moreover, as Boris the Spider pointed out: "... there is absolutely no guarantee that Random.nextInt() won't decide to return 1 for ever and ever. ..." Therefore, it might be possible that your code never terminates (although it will be very unlikely). You can make it faster in the following manner:

import java.util.Random;

public class Domain {

    public int[] randomIntArray(int size) {

        Random random = new Random();
        int array[] = new int[size];
        // populate the array with sequential values
        for (int i = 0; i < size; i++) {
            array[i] = i + 1;
        }
        // "shuffle" the array
        for (int i = 0; i < size; i++) {
            int j = random.nextInt(size - i) + i;
            // swap
            int temp = a[i];
            a[i] = a[j];
            a[j] = temp;
        }
        return array;
    }
}

It's running time is linear i.e. \$\Theta(n)\$¹. I am basically describing the Fisher-Yates shuffle

Some Background Info: The Big-O stuff

The Big-O notation is used to define an upper limit of the number of operations executed (the maximum running time, in simpler terms). For example, if the exact number of iterations based on a given parameter (like the number of elements, digits, etc) called \$n\$ is $$2n^2+3n-4$$ we can say that a reasonable upper limit is \$3n^2\$. In the Big-O notation, the constant term gets absorbed. When we represent this in the Big-O notation, we say: "the (maximum) running time is proportional to \$O(n^2)\$".

People tend to forget the other two notations: the Big-Omega (\$\Omega\$) and the Big-Theta (\$\Theta\$) The former defines a lower limit and the latter is the Big-O and the Big-Omega combined together. In the example above, the lower limit can be \$2n^2\$. Absorbing the constant, we say:"the (minimum) running time is proportional to \$\Omega(n^2)\$". The function can therefore be bounded within both upper and lower limits, or both the upper and the lower limits are in the order of \$\Theta(n^2)\$.

Your code has no upper limit. It only has a lower limit. So, only the \$\Omega\$ is defined for your code (unless we set out to do a lengthy mathematical analysis of the behaviour of the nextInt() function).

I suggest that you read the related Wikipedia articles. (I didn't cover \$o, \omega\$ and \$\theta\$, that's your homework). Remember, Google is your friend!

¹ Assuming that the nextInt() method takes \$O(1)\$ (i.e. constant) time to execute.