HI

  will you check this.....

   I have defined a page which shows multiselect picklist and two Arrow mark image .You can check this 

   This is my page

<apex:page controller="multiselect">
    <apex:form >
        <apex:panelGrid columns="3" id="abcd">
            <apex:selectList id="sel1" value="{!leftSelected}" multiselect="true" style="width:100px" size="5">
                <apex:selectOptions value="{!unSelectedvalues}" />
            </apex:selectList>
                <apex:panelGroup >
                    <br/>
                    <apex:image value="{!$Resource.multiselected}">
                        <apex:actionSupport event="onclick" action="{!selectclick}" reRender="abcd"/>
                    </apex:image>
                    <br/><br/>
                    <apex:image value="{!$Resource.multiselected}">
                        <apex:actionSupport event="onclick" action="{!unSelectclick}" reRender="abcd"/>
                    </apex:image>
                </apex:panelGroup>
            <apex:selectList id="sel2" value="{!rightSelected}" multiselect="true" style="width:100px" size="5">
                <apex:selectOptions value="{!SelectedValues}" />
            </apex:selectList>
        </apex:panelGrid>
    </apex:form>
</apex:page>

 And My controller is 

public class Multiselect {

    Set<String> originalvalues = new Set<String>{'Monday am','Wendsday pm','Friday neight','Tuesday am'};
    public List<string>leftSelected {get;set;}
    public List<string> rightSelected{get;set;}
    Set<String> leftValues = new Set<String>();
    Set<String> rightValues = new Set<String>();
    
    public multiselect(){
        leftSelected = new List<String>();
        rightSelected = new List<String>();
        leftValues.addAll(originalValues);
    }
    
    public PageReference selectclick(){
        rightSelected.clear();
        for(String s : leftSelected){
            
            leftValues.remove(s);
            //string spilting logic
            String st = s.substring(0,s.indexOf('y')+1);
            System.debug('##################SouvikTest#############'+st);
            rightValues.add(st);
        }
        return null;
    }
    
    public PageReference unSelectClick(){
        leftselected.clear();
        for(String s : rightSelected){
            rightValues.remove(s);
            leftValues.add(s);
        }
        return null;
    }

    public List<SelectOption> getunSelectedValues(){
        List<SelectOption> options = new List<SelectOption>();
        List<string> tempList = new List<String>();
        tempList.addAll(leftValues);
        tempList.sort();
        for(string s : tempList)
            options.add(new SelectOption(s,s));
        return options;
    }

    public List<SelectOption> getSelectedValues(){
        List<SelectOption> options1 = new List<SelectOption>();
        List<string> tempList = new List<String>();
        tempList.addAll(rightValues);
        tempList.sort();
        for(String s : tempList)
            options1.add(new SelectOption(s,s));
        return options1;
    }
}

 Did this post answers your questions ,If so please mark it solved so that others get benifited.

Thanks

asish