1. We did this cruise this past year (November 2024). We stayed at the Hyatt Place near the port. It did not have airport shuttle, but did have a shuttle over to Old Town. We took a taxi to the hotel. Pro tip-- if you use a taxi from the airport, get in the official taxi stand line. Independent taxi drivers will try to get your business, but charge more.
The Hyatt is walking distance (or can take hotel shuttle) to the T-Mobile District-- several restaurants there.
On a previous cruise from San Juan, we stayed in Old San Juan at the Casablanca hotel and enjoyed our brief stay there (very comfy bed).
2. I believe
DCL has port transportation from the Hilton Caribe? It's on the beach, not quite near OSJ (and more resorty/expensive than where we have stayed before).
From the Hyatt, it was a very short Uber ride to the port, less than $15.
3. Yes, the Pan American terminal. I believe the cruise terminal near OSJ is primarily just for day stops, not full embarkation.
4. We did a food tour with Spoon Tours and really enjoyed it. We used the hotel shuttle to get to OSJ, then Ubered back to hotel afterwards.
Hope that helps!