I would not rent a scooter, but if you want to see a bit of the island, rent a car. I don’t know how many hours you are in port-we were in GC for a month so we had nothing but time—but if you don’t want to sit on the beach, or go for a stingray city tour, it’s nice to drive to the less crowded part of the island. Rum Point is lovely. Sometimes it is very crowded. Kaibo is close by and is not crowded at all—but if you want to go in the water and snorkel, it’s better at Rum Point. Restaurant at Kaibo is nice and less crowded. On the way over, you could stop at the Botanic Garden, if that is something you might be interested in. I took an 11am Iguana tour there this year that I found interesting. It takes about 45 minutes to drive from the airport to Kaibo, so probably another 10 minutes or so from the harbor. Then you have to take into account the time to rent a car etc.
Have you checked in the forum about what to do in one day. I think there is a part for people who are in on a cruise ship.