[SalesForce] Get the Name of a related record

You can do something like this. Basically you would need to get the opportunities of all the records that you're inserting / updating, get the associated accounts and then set the account values in the customOject's account__c field.

trigger autoPopulateTrigger on customObject__c (before insert, before update){

    //get the oppotyunity Id's and store it in a set.
    set<Id> opptyIdSet = new set<Id>();
    for(customObject__c co: trigger.new){
        if(co.opportuntiy__c != null){
            opptyIdSet.add(co.opportunity__c);
        }
    }

    //query the opportunity records and get the associated accounts.
    map<id, opportunity> opptyMap = new map<id, opportunity>{[SELECT id, accountid from opportunity where Id IN: opptyIdSet]};

    //update the account value based on the opportunity in the record.
    for(customObject__c co: trigger.new){
        if(opptyMap.containsKey(co.opportunity__c)){
            co.Account__c = opptyMap.get(co.opportunity__c).accountId;
        }       
    }

}

You can use a workflow rule for this functionality. Create a workflow with the field update as the workflow action. As Account is a lookup on the case object, you can reference account object's field in the field update using the formula editor in the field update screen.