I have been to Innsbruck twice and stayed at two different hostels.
There is the HI hostel
which is a little bit out of town (but can get the bus there). It feels sterile and a bit like an instituition but is the cheapest option. The lockout period during the day can be a bit of a nuisance.
Alternatively, there is St Nikolaus
which is more central but more expensive. It's a real hit and miss afair. The rooms are a bit crammed and bathroom facilities not great. The owners are a bit strange including the old fat German guy (in his defence he drove us to a nightclub in the middle of the night!). They have an underground bar called the 'Iglu' which is cool to hang out in.