I would deal directly with the driver about a quote and make sure the driver is insured to carry passengers (red plate).

If you find yourself stranded at the airport for some reason, you can walk out into the JUTA lot and book a ride to Negril for approximately $20 - $25 per person one way. They will take you to your hotel. It is a shared ride so you will stop at other properties along the route. Airport staff will tell you where the lot is.

Plenty of flexible transportation options are available and negotiable. Fares will vary according to your needs; private charter vs. shared ride, number of people in the party, amount of luggage, destination.