Do you have a particular reason to be worrying? - if not then don't, cos it tends not to happen. But if it does, it depends whether you booked the flight as part of the cruise. If you booked a flight independently, Princess' agents in Mo'Bay will help you sort accommodation & flights to catch up with the ship - all at your (massive) expense. But probably you've booked the whole shebang as one package, flying from Gatwick on an aircraft chartered by Princess & full of Sea Princess passengers. Then it's Princess' problem - but no problem at all. If one of their flights is late, even by many hours, the ship will wait for it. If I read your itinerary correctly, the day after you sail is a day at sea, & the ship is due in Dom Rep the morning after that. So you'll all still be in Dom Rep on time, it just means the cap'n will have to press a little harder on the loud pedal. That's if the flight is delayed. Very different rules if you MISS the flight. So don't ! Have fun. Chris.
If you are on the full Princess package (flight & Cruise) you will not miss the ship. If you fly independant then go a day early. to be sure.