Both a friend and I have used Hotwire and Priceline for day rooms at one time or another. The further in advance you start looking, you might strike a deal. You can check in probably at noon or 2pm and stay until you need to go to the airport.
If you don't find a price you are willing to pay, there are several options for showers in Anchorage - Spenard Hostel allows non-guests to shower, one of the campgrounds does as well and there are at least two rec centers (one not far from the airport) that are quite cheap for entry and that I have used just for a shower.