Yes the bus is the best way. The slight drawback is that Dundee has two cruise terminals- the old one close to the city centre, and a deepwater terminal quite a way out of the city, I've been to both.
From the deepwater terminal you would need a ship's shuttle bus (or taxi) to the bus station.
At the old terminal that is at a working freight dock. So you take a silly little shuttle bus about 200 yards from the ship gangway to the dock gate from whence it's walkable (or a taxi) to the bus station. You can't walk on the dock itself. Repeat that on the way back in. When I've been there the shuttle has been run by the local sea scouts of all people, in their minibuses. The 'terminal' is a marquee.
You are on the Azamara Journey I see so a fairly small ship, berthed from 0545 to 0200 next day thus lots of time. The timings tell me (apart from the ship's size) that you are at the old terminal, as you can only berth and deberth there at certain tidal states.
The whole itinerary is a very interesting one.
The current return bus fare is £9.40- a North East Fife Day Rider (or £8.20 each way).