Dubrovnick is a wonderful port to visit.
Even if you are not of the Jewish faith, a visit to the Dubrovnick Synaygouge is very interesting because of its history and cultural significance (walk up the stairs and see where the women sat).
But the most interesting thing to do in Dubrovnivck is to walk on top of the old city's walls. This is an unbelievable travel experience because from the top of the walls you look out to the Adriatic and over the city.
Wear comfortable shoes when you visit the fascinating city. As for transportation you should check with the ship's staff when you get on the ship, because some cruise lines have shuttles. You can do your own research and see this city on your own.
Hope that this helps.