AKA 'Kunsthistorisches Museum'. Enormous (seriously, ENORMOUS) art collection amassed by the Habsburgs and housed in absurdly ornate surroundings. You could spend a week here and not get close to seeing everything.
Getting there: From the hostel, take the U3 U-Bahn line from Westbahnhof (direction 'Simmering') to Volkstheater. Or you can walk there in about 20 - 25 minutes.