You really shouldn't be initializing your StandardSetController in the getter method; this effectively causes it to re-initialize itself each time the page performs any action at all. Instead, move the assignment of the new StandardSetController to the setCaseStatusMethod, and call this function during your constructor to initialize the default listing. You can actually refactor this a bit better by changing your page as well. Here's my suggestion:

<apex:page sidebar="false" controller="MyIncidentsPageController">
<!-- removed script -->

<apex:form >
    <apex:outputPanel >
       <apex:selectRadio value="{!caseType}">
           <apex:selectOptions value="{!items}"/>
           <!-- moved onchange event to actionSupport -->
           <!-- added new function to refresh database -->
           <apex:actionSupport event="onchange" action="{!refreshCaseList}" />
       </apex:selectRadio> 
    </apex:outputPanel>
    <apex:outputPanel style="padding:10px;display:block;" id="caseList">
    <apex:dataTable value="{!Cases}" var="OI"  headerClass="header" style="border-style:solid;border-width:2px;border-color:#c8c8c8;" rowClasses="odd,even" width="100%">
        <apex:column headerValue="Incident" style="padding:10px;">  
                <apex:outputText >{!OI.CaseNumber}</apex:outputText>
        </apex:column>
        <apex:column value="{!OI.Created__c}" headerValue="Created" style="padding:10px;color:#2f627f;"/>
        <apex:column value="{!OI.Modified__c}" headerValue="Modified" style="padding:10px;color:#2f627f;"/>
        <apex:column value="{!OI.Priority}" headerValue="Priority" style="padding:10px;color:#2f627f;"/>
        <apex:column value="{!OI.Productweb__c}" headerValue="Product" style="padding:10px;color:#2f627f;"/>
        <apex:column value="{!OI.Submitted_By__c}" headerValue="Submitted By" style="padding:10px;color:#2f627f;"/>
        <apex:column value="{!OI.Subject}" headerValue="Subject" style="padding:10px;color:#2f627f;"/>
        <apex:column value="{!OI.Status}" headerValue="Status" style="padding:10px;color:#2f627f;"/>
    </apex:dataTable>
     <apex:panelGrid columns="3" style="float:right;" id="links">
    <apex:commandLink action="{!previous}" rendered="{!hasPrevious}" style="padding:12px;">Previous</apex:commandlink>
    <apex:outputLabel rendered="{!(hasPrevious && hasNext)}" >|</apex:outputLabel>
    <apex:commandLink action="{!next}" rendered="{!hasNext}" style="padding:12px;">Next</apex:commandlink>
    </apex:panelGrid><br/>
    </apex:outputPanel>
    <apex:outputpanel style="float:right;display:block;margin-right:7px;padding-top:2px;">

    </apex:outputPanel>
       </apex:form>
</apex:page>

public with sharing class MyIncidentsPageController {

    // Removed List<Case> caseList. It's redundant, and we don't need it.
    // Removed String caseStatus, and associated functions. We don't need it. 
    // Removed function setCaseStatusMethod. We don't need it.

    // We now use the default getters and setters instead of getcaseType and setcaseType.
    public String caseType { get; set; }

    // Use default getters and setters instead of initializing controller here.
    public ApexPages.StandardSetController con { get; set; }

    // Single function that initializes the case list.
    public void refreshCaseList() {
        con = new ApexPages.StandardSetController(Database.getQueryLocator([select Id,Subject,CaseNumber, Resolution__c,AccountId, Status, Priority,Productweb__c, Created__c,Modified__c,Submitted_By__c,CreatedDate from Case where status=:caseType order by CreatedDate desc]));
        con.setPageSize(10);
    }

    public MyIncidentsPageController(){
        caseType='Open';
        // We initialize the StandardSetController here
        refreshCaseList();
    }
    
    public List<SelectOption> getItems(){
        List<SelectOption> options = new List<SelectOption>();
        options.add(new SelectOption('Open','Open Incidents'));
        options.add(new SelectOption('Closed','Closed Incidents'));
        return options;
    }

    // Changed this function to a getter only (it should be a read-only property).
    public Boolean getHasNext() {
        return con.getHasNext();
    }
    
    // Changed this function to a getter only (it should be a read-only property).
    public Boolean getHasPrevious() {
        return con.getHasPrevious();
    }
    
    public void next(){
        con.next();
    }

    public void previous(){
        con.previous();
    }
    
    public List<Case> getCases(){
        // Just return the list from the StandardSetController.
        return con.getRecords();
    }

}

This clears up some extra variables (which will reduce view state size, which will reduce page load times), and causes the StandardSetController to only be initialized as necessary. This code is still overly complex, because the StandardSetController's methods can be called directly from Visualforce, so the getters, setters, and functions here are redundant.

When a page is loaded the first time, the Constructor is called, then any getter methods are called. So, by initializing the StandardSetController in the constructor, we will ensure that con is not null before it is accessed the first time. We changed your onchange javascript attribute to an equivelant actionSupport Visualforce tag, which is Visualforce's way of automatically generating whatever JavaScript needs to be created for an element's events on the fly. This way, we eliminate the need for an actionFunction and the extra functions and variables that you had to support this framework.