1. The usual route for young people to get a work visa in Japan or Korea is to teach English, but many schools will not hire anyone who is not a native speaker.

2. Many universities in Japan have programs for foreign students in which you take Japanese language classes and then other courses in English. A few universities in Korea also have these kinds of programs. In Japan, International Christian University (ICU), a bilingual university in the suburbs of Tokyo, has the best-known such program. (You don't have to be Christian to enroll. Most of the students are not.)

If there is a specific field that you want to study, then you may consider applying for a Japanese government graduate fellowship. This includes language training and time to study for entrance exams.

3. Your ability to get a work visa depends on whether you have a skill that Japanese or Korean employers need.

4. The airfare that you cite is a good deal. The cheapest way to travel within Japan is with a Japan Rail Pass (you have to buy the voucher for it outside of Japan) and since you are only 24 years old, youth hostels may be perfect for you. Japan as a lot of them, but I don't know about Korea.

Depending on your country of citizenship, you may qualify for a Working Holiday Visa to Japan or South Korea. You obtain the visa from the Japanese or Korean embassy in your home country, and then you are automatically authorized to seek any legal job in the host country. Typically, the visas are for six months, renewable once.