ok The adventure of the seas do not dock near old San Juan for a few reasons:
1. The Adventure of the seas is so big it cannot dock on the old San Juan side harbor. Actually she is using the old Carnival Destiny dock, and thats part of the dock mostly used for importation merchandise to the island.
2. When the Carnival Destiny arrived to San Juan was also so big they have to make some space for the ship on the San Juan bay, at that time our only "Cruises port" and since Carnival pay good money for it... is like their dock now.
As for how far is from old San Juan?
thats a problem you will have to take taxi is a little far, but also is a better dock than the one Carnival Destiny uses and is way less people boarding and is easier to board ship...