"Does this city bus go to the post office?"
Translation:Ця маршрутка їде на пошту?
If it's about you using a vehicle (I go by bus) - not acceptable at all. However, it can be used to speak about route of some transport.
Цей автобус іде до вокзалу. (This bus goes to the train station).
However, I would say that word їхати in this situation is still more common.