I'm confused, too -- are you going to visit South Korea? If so, I would strongly recommend the hostel in which I stayed in Seoul and, depending on your interests in Busan, the budget hotel in which I stayed there (my interests were a bit different than thursdaysd's), each of which is described in the trip report I linked upthread.

In Taipei, I was would have a look at http://www.cityinn.com.tw/Mobile/ they have sites all owner the city. On near to Taipei Main Station would make it a lot easier to get around using the subway. Not sure if you are going to Hualien, but we really liked Holland Windmill B&B there.

In Tokyo http://www.andon.co.jp would be worth a look.

In addition to AirBnB ( we have had mixed results there) www.hostelworld.com is more geared up to your age group.

Oh oops, I just completely forgot to add South-Korea to the list! I was a bit tired yesterday, haha.
I meant the 4th of August, yes, and of course I bought a plane ticket to South-Korea as well.
I will be going there from the 4th of August till the 13th of August.

Thanks for the help! I will take a look at the links you all provided.