[SalesForce] how to update picklist field according to another object

As I understand you want to populate picklist values in objects names? You can do this only by using Visual Force picklist. Of course you can put object name in picklist using trigger, because picklist is just kind of text field. But you just put there just some string, availble values are still same as before this update.

What you should do is to create VF page (1 if this field is read-only for user - just to show value, and 2 if user should be able to change value - in that case you have to create also VF page for edit)

Here is some example of the code:

public class Object1picklistcontroller{

Object__1 obj;

public Object1picklistcontroller(ApexPages.StandardController controller) {
    obj = (object1) controller.getRecord();
}

public SelectOption[] getOptions() {
    SelectOption[] options = new SelectOption[] {};

    for (Object__2 obj2 : [SELECT Id,Name FROM Object__2 WHERE Object__1 = : obj.Id]) {
        options.add(new SelectOption(obj2.Id, obj2.Name));
    }
    if (options.isEmpty()) {
        options.add(new SelectOption('', '--None--'));
    }
    return options;
  }
}

Page picklist

<apex:selectList value="{!Obj.Object_2_Id__c}" label="Type" size="1">
    <apex:selectOptions value="{!Options}"/>
</apex:selectList>

First, you never want to put a query inside of a For loop. As it stands right now, you're querying for each record when you could have made a collection first and then run one single query for all of the related records in obect2__c.

That said, since the record Id's in object2__c are the same as the Case Id's (from your query, object2.Id =: cs.id), you don't even need to run a query at all. Instead, all you need to do is determine whether or not your picklist values have changed, then update the related records using the same Id's.

Finally, this is an After Update trigger, not both a Before and an After update. a Before Update runs before the updates on Case are committed to the database. You don't want to update the related object until the updates have been written to the Case Object and you only want to write them once.

Here's your revised trigger:

trigger casetoupdate on Case (after update) {

   List<object2__c> toUpdate = new list<object2__c>();

   for(Case cs : Trigger.New) {     

      // compare new and prev picklist values to create a list of Object__2 records to update

      if(cs.picklist__c != trigger.oldMap.get(cs.Id).picklist__c) {

         object2__c toUp = new object2__c(Id = cs.Id, picklist__c = cs.picklist__c); 

         toUpdate.add(toUp);

      } // end of if statement

   } // end of for loop

   if(toUpdate.isEmpty() == false) Update toUpdate;

} // end of trigger