Are you staying one night in Dubrovnik or just doing a day trip from the ship? If just a day trip, how many hours?
We we just there in April for three nights and the crowds from the cruise ships would come into the old town area around 9:00 AM and depart around 1:00 PM. In fact, a lot of the shops set their hours to coincide with cruise ship visitors.
Let us know how many hours you have and we can provide some suggestions, beyond the obvious one of walking around the top of the city walls.