I've noticed that no one has responded yet. The Fairbanks folks will probably catch up in a day or so and while I don't have the specific FAI knowledge, I'll pass some Alaska generalities on to you.

There are a number of hotels within a couple of miles of the airport and the hotels are used to arrivals at all hours. While their web site may say they they have shuttle service, it may be limited hours. I've also found that some hotels use a contract service rather than their own vehicles and employees.

I've always been successful in getting the information with a phone call. You can get a list at www,hotels.com and it's only a few miles from FAI to downtown.