I'm sure you could find a short term lease at any APT complex. Anchorage is really no difference than the lower 48 when it comes to that stuff.
As for the bus - it is great if you have to use it. You may not want to stand in the cold too long, but they run great in the Winter. Downtown bus stops are mostly covered. Mid town and out are ususaly just sign posts so you will be standing in snow waiting.
What part of town is work going to be? I could help you little on the areas.
Good Luck
Oh yeah! I had a 6 month lease on a fully furnished apartment. Not a bad place. Just the basics. Included Cable (But no TV) and everything. All you had to pay for was Phone if you even got one. I just used my Cell. That was abpout 5-6 years ago so I can't tell you what the cost is now. Good central location too. I can let you know more about it if you interested.