If you want make like a local and live in a residential setting,
Caleta Balcony Rentals (11 Caleta de las Monjas, 787/725-5347, fax 787/977-5642, reservations [at] thecaleta [dot] com, www.thecaleta.com/guesthouse.html [1], $80–150) offers six cozy, modest units varying from studios to one-bedroom apartments in a three-story structure near the San Juan Gate. The property is a bit shabby, but it has a great location between Hotel El Convento [2] and La Fortaleza [3], home of the governor.
All apartments have balconies and a full kitchen or kitchenette; some have air-conditioning, TV, and telephones. Each room is different, and they all have character to spare, but the Sunshine Suite on the third floor is the best of the bunch. There’s no reception desk or on-site management at this self-serve property.
Links:
[1] http://www.thecaleta.com/guesthouse.html
[2] http://www.moon.com/destinations/puerto-rico/san-juan/accommodations/old-san-juan/over-250
[3] http://www.moon.com/destinations/puerto-rico/san-juan/sights/old-san-juan/la-fortaleza