If you don't mind night buses, there are a couple of express buses that depart from Chetumal (at 10 pm and 12:30 am) and go direct to the airport. It's a 5.5 hour trip so the 10pm bus will get you there with plenty of time to spare. The 12:30 am bus will get you there at 6am (probably too close for comfort).
I usually stay in Playa, but with a 7am departure time, I might not advise this for you because Playa is about 1 hour south of the airport and the airport express buses (110 pesos) start running at 5am (getting you to the airport at 6am). You would have to leave Playa by 3 or 4am. There are probably other ways to get to the airport but I don't have any details about them. If you were forced to take a taxi, it would be more expensive from Playa than Cancun city.
Cancun City is just under 30 minutes north of the airport, and there are frequent buses to the airport. There are a few hotels near the bus terminal -- Hotel Alux (I think it's a lower budget hotel) gets good comments on forums (I've never stayed there).