I've been to two hostels in Budapest in 2001 when I wasn't able to secure a hotel room, one was really crappy, and the other one was very cozy and neat.
The good one was very very close to the Nyugati train station, the one next to a big shopping mall. However, it was normally used as a dormitory to a university or something, so it might be unavailable as a hostel this time of the year and I can't seem to locate it on google maps...
In 2005 (when we've been to Budapest for the QPR concert!) me and my friends all stayed at the Corinthia Hotel, which was pretty good.
So, in short, there are some pretty good and pretty crappy hostels out there in Budapest, you can't be certain which one's good until you check in... I'd recommed Corinthia if you have the means as it's quite fail-safe :)
Get on your bikes and ride!