The Princess would not be my first choice for a place to stay in BZ City but if it's only for one night then hopefully you'll survive, lol. That place gives me the heebie jeebies. It's got a casino on site and is full of mirrors and other cheap glitzy stuff. And there's a weird, kind of musty smell to it.
If it's not too late, you might want to check on either the Ft George Radisson or, my preference, The Great House right across the street. Both are a five minute cab ride to the Princess Marina.
No matter where you stay, don't go out at night. BZ City is a rough town, to put it mildly.
As far as daylight goes, you have lots of choices. You can walk around on the main streets in relative safety. A good walking tour would be to walk south along the seawall to the Baron Bliss Monument and on into the city. Check out the Swinging Bridge. There's a dive shop right before you get there. Cross the Swinging Bridge and continue south on the main drag until you get to St Johns Cathedral, about four or five blocks down. If it's open, check it out. Then go across the street to the Governor's Mansion and follow that street back into town. Other places to see along the way would be the Supreme Court Building, the Belize Museum and the old American Consulate.
Don't venture down any of the side streets, at least not very far. Don't wear expensive jewelry. Be aware of your surroundings. You may get pestered by beggars. Just ignore them and they'll go away. And you may get offers to sell you marijuana. It's illegal there so partake at your own risk. Hattieville Prison is not a place you want to spend your vacation.
Or you could arrange to go cave tubing and/or zip lining (same place). You'll need a guide with a car, which is easy to line up and your hotel will do that for you.
Or you could check out the Mayan ruins at Altun Ha. Takes about an hour to walk around. Again, you'll need a car and a guide.
Both of those are close enough that you would have plenty of time to do them and be back in time for the boat. Plus, you'd get to see a little more of Belize, although not the prettiest parts.
Have a great trip!