If the adventure is still sailing at 10 PM and you are there for early check in, then yes, you'll have time to go site seeing. You may just want to check in your bags and have your cab wait for you rather than fully checking in. You can be back around 5 pm and complete your check in then. Saves you from waiting in a long line up.
Just be advised, the Adventure of the Seas boards at the Pan American Pier, not at the numbered piers in Old San Juan. You will have to get a taxi to take you to Old San Juan and then back to the ship.
Majesty of the Seas - 10/03 & 02/07 & 11/08
Enchantment of the Seas - 10/04 & 10/11
Mariner of the Seas - 10/05
Vision of the Seas - 10/06 & 09/07
Carnival Liberty - 10/07
Adventure of the Seas - 9/08
Ruby Princess - 12/09
Voyager of the Seas - 9/10
Carnival Spirit - 11/10
Oasis of the Seas - 9/13