First, the Senate needs to expel him. That takes a two-thirds vote, which will happen as soon as he takes office.
Once he is expelled (both parties have agreed to do so in the event of his victory), then Palin will appoint a Republican of her choosing to fill his seat until a special election is held.
In Alaska, a special election must be held within 90 days of his resignation/impeachment. However, at that point, it's anyone's game and up to the party. It's essentially like having a fresh election, except it'll just be for the senate. So Begich will have to run all over again, if he chooses.
Palin cannot appoint herself as the 90 day replacement, but she is allowed to run in the special election.