You can get taxi's at the gate entrance to the Port area to take you into town. They will stay with you and bring you back when you're finished. Negotiate a rate first and they can be great... lots of interesting commentary from them too. They will also try to take you to one of the remarkable jewelry shops in Cartagena... great buys in gold and Emeralds... and other precious stones. I wouldn't suggest trying to do it on your "own". There is no bus service from the closed off pier area into downtown... except occasionally for the locals.
If you are not comfortable in a basically non tourist area, you may well be better off taking a ship's tour at this Port. It is great and interesting, but the tourist infrastructure is not what you might expect.