One of the Must-Sees in Vienna - A Thirteenth Century former imperial palace, most famously the Winter residence of the Habsburgs. Wander around outside and admire the obscene grandeur for free, or do a tour of the Imperial Apartments, the Sisi Museum and the Silver Collection (Audio guide €12.90, human guide €15.90. Students get a whole Euro off). It's a beautiful place to wander through in the evening, too, when it's all lit up.
Getting there: From the hostel, take the U3 U-Bahn line from Westbahnhof Station (direction 'Simmering') to Herrengasse. It's a short walk from there.