Definitely rent a car- this time of year you ought to be able to get a pretty good deal.
The aurora, when it's visible, isn't usually seen from downtown- too many lights. Your best bet is to get out of town if there's a cloudless night and hope for the best.
If you ski, either downhill or X-C, there are lots of opportunities. Cross-country is especially good here, the Chugach Sate Park, "Anchorage's Back Yard", has almost a half-million acres to explore. For a great view of the Anchorage Bowl and the surrounding coastline and mountain ranges, visit the Glen Alps parking lot at the base of Flattop Mountain. Spectacular views, and also a good place to see the northern lights if they're out.
As far as other sights during your very brief stay, we've got a couple of good museums, but they'll probably be closed on Christmas. (None of my business, but that on earth prompts such an untimely visit?)
Hope this helps. If there's anything else I can help with drop me a note at [email protected]. Good luck with the trip plans.