Fromflorida...a day trip can be done, although it will be a very long day...you seem to understand this issue.
I also agree with Bigbamboo...make sure the driver knows Kingston, as it can be easy to get lost there and will making the drive annoyingly longer!
You can email a few drivers from the transportation section of Negril.com https://negril.com/transport/ and ask for quotes. Don't be surprised if some quotes are around $500us for the return trip.
I wouldn't plan much more for the day trip other than Port Royal (which is a fabulous visit) other than maybe getting a delicious meal at Gloria's in the town of Port Royal.