Yes, a taxi would work, Uber is also available; a few hotels have shuttles. There is a public bus out to the airport, but not that early. A flight that early would be on a plane that spent the night there, so that a late incoming flight would not be the source of any delay. Fairbanks actually has an enviable record for on-time departures relative to other airports. The flight density is not high, and when the weather is bad, they are used to dealing with it.