Having enough vacant apartments has posed a real challenge for Anchorage over the last several years. Because of the climate here, you can often have good success searching for an apartment in the winter, as some people will sublet their homes to live in better climates for the winter months. If you start searching at the end of one winter, you may also have good luck booking into a place for the following fall.
As in any market where there is low rental apartment availability, you need to think of every rental application you submit as a competition. Do anything you can to make yourself stand out from your competitors. For instance, showing up with letters of reference in hand before you even apply can show that you take getting this apartment seriously, and that you have the references to prove that you're worth renting to.
Anchorage is very spread out, due in part to the rough terrain that it is built on. While it is possible to cycle throughout the city, walking is very difficult unless you don't need to leave the downtown area. The PeopleMover bus system is helpful, but has infrequent routes. Driving by car is the best way to get around, but if you're moving from down south you will need to invest in a set of winter tires for the colder months.