I have not stayed at HI Perugia but have lived in the city for quite some time so I might be able to answer some of your questions:
1) Yes, both the main train station (Fontivegge) and the HI are outside the historic centre (which sits on top of several hills).
2) To get from the station to the hotel you could use public transport, it will be a hassle though especially if you travel with luggage: Bus line F (direction "San Sisto"/"Silvestrini") leaves from near the station and should take you to the main road near the hotel (Via Firenze, Strada Trasimeno Ovest). From there it should be a 5 min walk.
Rental car companies with offices at the train station are: AVIS, Hertz and Sixt.
Taxis are available in front of the station. The price should be somewhere around 15 EUR, depending on the traffic.
According to the hotel's website there is a shuttle service upon request (however, neither the English nor the Italian website states whether the service is free of charge). This would probably be your best choice.
3) According to the hotel's website they offer a scheduled shuttle service (free of charge) to the city centre at 9 AM, 3 PM and 6 PM. At other times on request. They will probably drop you off at Piazza Italia (my guess) which would be the perfect starting point to explore the town.
4) There is one other ICHG property in Umbria, the HI Express in Foligno. Foligno is about 40 km or a 45 min train ride from Perugia and not a very interesting town.
If you have any other questions about Perugia or Umbria, feel free to ask!