We went to San Juan on the Sky Princess earlier this year. Five ships in that day (!), four of which including us docked very centrally. One Holland America at the more out of town Pan American.
If you docked where we did, you'll be right in the Old Town. Key sights walkable. The National Parks service runs two incredible historic fort sites. To get to them, you can't help but walk through the historic old town and along the coast. Stunning views, lots of Spanish influence, and interesting history.
No taxi, organised excursions etc needed. Entry to the two forts was very reasonable, $10 for both I think. A few photos. Enjoy your day!