Although I said its good have relationships, its not impossible to acheive using just the names.

I have modified the regular code to handle names. Its still not complete as it needs some work to be done, but maybe you can start off with this.

trigger RollUp on Resume_Subscriptions__c (after insert, after update, before delete) {

    // Set to hold the IDs of the Contacts of the Subscriptions
    Set<String> ContactNames = new Set<String>();
    // Set to hold the IDs of the Subscriptions
    Set<ID> SubIDs = new Set<ID>();
    // Set to hold the IDs of the Contacts where the values need to be reset
    Set<ID> ContactNamesForReset = new Set<ID>();

    if(trigger.isupdate || trigger.isInsert){
        for(Resume_Subscriptions__c rsSub : trigger.new ){
            // If the contact of a subscription has changed, then the old contact also needs to be updated 
            if(trigger.isUpdate)
				if( rsSub.User__c != trigger.oldmap.get(rsSub.ID).User__c)
					if(trigger.oldmap.get(rsSub.ID).User__c != null )
						ContactNames.add(trigger.oldmap.get(rsSub.ID).User__c);
            if(rsSub.User__c!=null){
                ContactNames.add(rsSub.User__c);
            }
        }
        ContactNamesForReset.addAll(ContactNames);

        // The list of Contacts where the update has to take place
        List<contact> ContactsToUpdate = new List<contact>();    

        // Querying the database to get the Number of Subscriptions under an contact and the sum of the deal values of the Subscriptions under an contact
        AggregateResult[] Subscriptionsum = [select count(ID) NoOfSubscriptions, User__c ContactName from Resume_Subscriptions__c where Active__c = true AND User__c in: ContactNames group by User__c];

        for( AggregateResult ContactDetail : Subscriptionsum ){
            contact ContactToUpdate = new contact(Name = (String)(ContactDetail.get('ContactName')));
            ContactToUpdate.Active_Resumes__c = integer.valueOf(ContactDetail.get('NoOfSubscriptions'));
            ContactsToUpdate.add(ContactToUpdate);
            // The contact being handled should not be reset. Hence removing it from the ResetSet
            ContactNamesForReset.remove((String)(ContactDetail.get('ContactName')));
        }
        
        for( String nm : ContactNamesForReset ){
            // Resetting the summed up values to 0 if the contact no longer has a subscription.
            contact ContactToUpdate = new contact( name = nm );
            ContactToUpdate.Active_Resumes__c = 0;
            // ContactsToUpdate.add(ContactToUpdate);
        }
        
		// some more logic is needed to identify these records.
        // if(ContactsToUpdate.size() > 0)
            // update ContactsToUpdate;

		ContactsToUpdate.clear();
    }

    if(trigger.isdelete){
        for(Resume_Subscriptions__c rsSub : trigger.old ){
            if( rsSub.User__c != null)
                ContactNames.add(rsSub.User__c);
            SubIDs.add(rsSub.ID);
        }
        ContactNamesForReset.addAll(ContactNames);

        // The list of Contacts where the update has to take place
        List<contact> ContactsToUpdate = new List<contact>();    
        
        // Querying the database to get the Number of Subscriptions under an contact excluding the Subscriptions being deleted
        AggregateResult[] Subscriptionsum = null;
        if(ContactNames.size() > 0){
            Subscriptionsum = [select count(ID) NoOfSubscriptions, User__c ContactID from Resume_Subscriptions__c where User__c in: ContactNames and ID not in: SubIDs group by User__c ];

            for( AggregateResult ContactDetail : Subscriptionsum ){
                contact ContactToUpdate = new contact(Name = (String)(ContactDetail.get('ContactID')));
                ContactToUpdate.Active_Resumes__c = integer.valueOf(ContactDetail.get('NoOfSubscriptions'));
                ContactsToUpdate.add(ContactToUpdate);
                // The contact being handled should not be reset. Hence removing it from the ResetSet
                ContactNamesForReset.remove((String)(ContactDetail.get('ContactID')));
            }

            for( String nm : ContactNamesForReset ){
                // Resetting the summed up values to 0 if the contact no longer has a subscription.
                contact ContactToUpdate = new contact( Name = nm );
                ContactToUpdate.Active_Resumes__c = 0;
                ContactsToUpdate.add(ContactToUpdate);
            }
        }

		// some more logic is needed to identify these records.
        // if(ContactsToUpdate.size() > 0)
            // update ContactsToUpdate;

        ContactsToUpdate.clear();
    }
}