If you just need a place to stay in Anchorage for one night before you go to Homer, you might consider the youth hostel near the airport. It is called the Spenard Hostel International. Three of us stayed there when our flight arrived at Anchorage around 9:00 PM and we were heading to Seward really early the next morning. We paid extra to have a room for the three of us. A bed in a shared room with a shared bath is $18 cash or $19 with a credit card; we paid $76 for a private room with two bunk beds; one of the bunks stayed vacant.

We flew first class to Anchorage (on miles) and then went to the hostel. My Louis Vuitton luggage got some strange looks when we checked in; however, the room was clean and quiet; the bunk beds had gorgeous hand-made quilts.
We did have to pay $0.25 for each towel we wanted to use. The bathrooms were clean and we did not have to wait. There is also a shared kitchen if you want to fix a snack or breakfast (bring your own food).

The hostel is located on Spenard and is near the airport. They also allow non-guests to rent their bathroom to shower and freshen up for $3.00. The Anchorage airport did not have public shower facilities when I was there in September 2004.

Their website is www.alaskahostel.org