Old San Juan teems with the rich cultural history of Puerto Rico, and it should be part of anyone's journey to this island.
The country has done a great job of maintaining the old fort, El Morro, and the walls that guard the old city. You can walk from the fort down to the bay area (port side) just by following the outer wall. It's worth the walk to see the beauty and envision the past.
Ask any local to tell you stories and histories of the old city. You can even grab an apartment for the night downtown and sleep in the old architecture with indoor courtyards and balconies covered in plants. There are great spots for drinks and eats—wander the streets and try everything.