Fairbanks in the middle of winter--are you going for work reasons and looking for off duty activities? You won't have much (if any) sunlight before ot after work--maybe none depending on your work hours and whether you are there in early or late January. If you aren't there for work, then you will have 4-6 hours of sunlight (on clear days) depending on what part of the month you are there.
Anyway, there are various winter tours from Fairbanks, including dogsled tours and others to see the Northern Lights. Experiences like ice fishing and the Chena Hot Springs are also available that time of the year. Alaska is not inexpensive, so tours can be pricey, but many are one-of-a-kind experiences.
I did live in Alaska for a few years some time ago--and I would suggest that, if you are going to book local tours/experiences, you try to speak with someone at the tour company before booking. Alaska does tend to attract some people with "unusual" personalities that, on occasion, can be a bit offputting and make what sounds like a wonderful experience, less so. Speaking with someone is one way to help avoid a less than ideal experience.