The weather in Anchorage is out of the ordinary warm right now. Still no snow on the ground in Anchorage. With such a short amount of time, not sure if flying to Fairbanks will be the best use of your time. As already recommended, Talkeetna is a great place to visit. You may also see the Aurora from there as you are far enough away from Anchorage's light pollution.

Another idea is to go see the Matanuska Glacier. It is about 1 1/2 hours drive from ANC. The Glacier is accessible enough that you can walk out onto the glacier a ways.

Anchorage has a decent museum. There are also several micro brewerys in town that make great beer if you are into that.