I live in Southampton so can tell you the car hire places are near the cruise terminal, they're within walking distance if you don't mind a bit of a hike along a busy road.
If all the places are closed on a bank holiday, then you might well be stuck with getting a car from the airport which is a taxi ride away. I have no idea what it would cost you as I have never taken a taxi anywhere in Southampton, but it's about 10 miles.
As for returning a car to Gatwick, that isn't a problem, but you'll often pay a premium for hiring a car from one location and returning it to another one. WIthout knowing which company you're going to use I can't tell you how much extra time to allow as I don't know how far from the airport you'll be, although most of the big car rental companies have an office in the terminal, but it all depends on where you have to leave the car in the first place.