I've taken the train from SEgovia and you should check the high speed train schedule because it's 25 minutes!
The bus company is La Sepulvedana, and they have a web site with schedules.
I don't know if there are buses to Segovia from the airport (I kind of doubt it). La Sepulvedana uses the bus station at Principe Pio. You can get to Principe Pio direct on the Cercanias line of RENFE and it takes about 45 minutes. If you're arriving in T4, that will be very easy, if you arrive in T 1,, or 3, you will have to take the airport bus that shuttles people around the terminals.
I haven't heard anything about the albergue in Ane being closed but maybe I missed something. I have only heard that it is very dirty and not maintained. But maybe that has changed as well. I didn't stay there because there are very nice alternatives if you schedule it right! Buen camino, you will love this route. Laurie