Housing in the Innsbruck area is tight and expensive. You are well advised to pursue all options: student dorms, shared apartments (Wohngemeinschaft, WG), and small apartments (here often known as Garconniere).
The best time to look for housing is during the summer months, since many students leave or arrive during time.
Many landlords will ask for long leases; between 1 and 3 years is common. The risk is limited though: Due to the high demand, it is usually not a problem to find a successor if you have to move out early. In case of doubt, ask the landlord about the conditions that apply in this case.
Sometimes students sublet their apartment or room for a semester they spend abroad. Such an interim solution can be a good option because it gives you time to get to know the city and keep looking for a good, long-term solution.