There's an easy answer if you are staying downtown...the Anchorage Museum at Rasmuson Center (6th and C, about a block from the Trooper Museum that you mentioned). There are also some good art galleries on G Street between 3rd and 5th if you are into that kind of thing.
Other than that, downtown Anchorage is pretty quiet this time of year, at least from a tourism perspective.
The Public Lands Information Center, as noted above, is also an interesting place to spend some time.