Add Radio Button to Custom Object

For this to work you'll need to commit the selections. Currently your page just displays the radio buttons, but changing the values doesn't go to the database. For this you'll need to use the save method.

<apex:page standardController="Pre_propo__c">
    <apex:form >
        <apex:commandButton value="Save" action="{!save}"/>
        <apex:selectRadio value="{!Pre_propo__c.Research_Cluster_2__c}">
            <apex:selectOption itemLabel="Public" itemValue="Public"/>
            <apex:selectOption itemLabel="Private" itemValue="Private"/>    
            <apex:selectOption itemLabel="Subsidiary" itemValue="Subsidiary"/>    
            <apex:selectOption itemLabel="Other" itemValue="Other"/>        
        </apex:selectRadio>
    </apex:form>
</apex:page>

That said, what you're going for - replacing a standard input field with a modified input method - isn't easy. If you continue with your approach you'll be close, but it means that a user will A) only be able to see/use the radio button on the "view" page and B) will have to remember to click the save button in the inline visualforce page (not the standard save button) to store any changes. You can override the new and edit pages with your own page, but then you have to create the rest of the layout as well.

One option would be to make that field read only on the page layout. Then add a button linked to your current page called "Change Research Cluster". The user clicks the button, changes the research cluster, clicks save, and goes back to the Pre Propo record. Not ideal, but less error prone.

In any event, the requirement, display a picklist as a radio button, may not be worth the effort.