Timing of your trip is pretty good as the tourist season ends in mid-September, and prices for hotels, etc., drop drastically while availability improves. In your price range you could try the Millenium Inn on Spenard, very close to the airport.

Anchorage is a spread out place with limited public transit. Getting around without a car is going to be a challenge. You can cab it of course but that'll get expensive in a hurry.

There are plenty of good eats in Anchorage. My personal favorite is the Beartooth Grill. It's a long while since I've been there but I recall that the blackened halibut tacos were excellent.