Leaving the airport & heading to Negril won't be a problem...keep your printed itineraries with you for the driver in the event you're stopped.
I don't know where you're staying...but you can (as you said) pick up food in Montego Bay or anywhere along the way into Negril.
Another option is to have a restaurant in Negril deliver food...maybe your hotel can help arrange this?
Don't forget about snacks & beverages for the evening...either pick those up enroute to your hotel or even easier...ask your driver to grab you a few items prior to your arrival.