[Hatredcopter]

In both Kyoto and Osaka there are hostels run by "J-Hoppers". I've never stayed at one, but I've been recommended them by multiple people. I believe they have a website you can look up.

However, I *have* stayed at the following places:
Budget Inn in Kyoto - the name is rather unspectacular, but it's a nice little hostel run by English-speaking staff, and you can stay for less than 3000 yen. They let you choose between private rooms and dorm-style rooms, as well.

Guesthouse U-En in Osaka - another hostel run by several friendly staff members, and they have a bar! I stayed for barely even 2000 yen if I remember right.

Both those hostels are very close to major train stations (Kyoto station and one of the Osaka loop stations, respectively.)