What are you specifically looking to update? Basically, updating an activity is just like updating any other entity, you just have to use the task
entity.
public void CloseTask(CrmService crmsvc, Guid activityid, DateTime start, DateTime end)
{
ColumnSet cols = new ColumnSet();
cols.Attributes = new string[] { "activityid", "statecode" };
task tsk = (task)crmsvc.Retrieve(EntityName.task.ToString(), activityid, cols);
if(tsk.statecode.Value != TaskState.Open)
return;
tsk.actualstart = new CRMDateTime();
tsk.actualstart.value = start.ToString();
tsk.actualend = new CRMDateTime();
tsk.actualend.value = end.ToString();
crmsvc.Update(tsk);
SetStateTaskRequest state = new SetStateTaskRequest();
state.EntityId = activityid;
state.TaskState = TaskState.Completed;
state.TaskStatus = -1; // Let MS CRM decide this property;
SetStateTaskResponse stateSet = (SetStateTaskResponse)crmsvc.Execute(state);
}
Edit: added some sample code. note, I had to modify what I had to strip some proprietary code, so I don't know if this will actually compile. It's close though.