You have various options. There is no public bus per say. Rather you could do route taxis which are extremely inexpensive but you'd need to get to the bus park in downtown MoBay and if you haven't done it before then I don't recommend it because you have to change buses and get on another in another bus park along the way in Lucea. Also it only works if you're traveling light luggage wise. The ride is cramped and uncomfortable and not really for tourists but some seasoned ones and ex-pats use it.
Next most inexpensive option is Knutsford. Their rates just recently went up but I think it's like $24/person last time I checked. Nice big charter bus with a/c, tv, wifi, and bathrooms. Your flight has to get in on time to utilize it though as it leaves MBJ daily to Negril just before 2PM. Downfall with this method other than needing to schedule your flight to its departure time is that it drops you at one centralized spot in Negril that has moved down towards the a/i's in the Hanover part of Negril so it requires catching a taxi from there to your hotel. They have a website and you can book it online.
JUTA is the other option and is also relatively cheap ($30 USD per person)but they generally don't leave the airport until they fill a van up and then it's a direct stop to the hotel with no stops. So if you need frequent bathroom breaks or like to make a stop or two along the way then it's not a good option. You can also book this online or just go up to the window at the airport near the departure gate and get one.
It's not hard to find a taxi at the airport but as mentioned they're supposed to be red plate licensed. Most of the guys that approach you in uniforms right as you exit are but occasionally you get a free lancer in there who isn't. You'll need to negotiate price but the going rate is $70 USD/person however if you have a group you can get the per person amount to slightly drop with some drivers. I have a favorite longtime driver named Vern 1(876)862-2677 he is red plate licensed. I've known him since 2000. I usually book with him, grab a beer at the bar outside the departure gate and hit him up on Whatsapp when I arrive.