[Firebird]

There are some airlines flying around Japan and Korea, but they are usually not as cheap as Easyjet or Ryainair.

I dont have alot of information about the airlines, but im sure you will find something for your first 2 flights.
From Korea to Japan it will maybe be more expensive or harder to find a cheap flight. If you have a bit more time, here is what i did: There is a Korea/Japan through Railpass wich works as following. You will choose your travel date and the starting city in Korea (Seoul for example). Than you choose your final destination in Japan (Osaka for example) and they will give u a price. This price will include the KTX train (Koreas Shinkansen) to Busan, the fast boat (3hours) to Fukuoka and the train (i think you have to pay extra for the Shinkansen, but you can use the Limited Express trains in Japan) to your destination in Japan. Their is a discount on all the trains and the boat, so it will be cheaper than flying. The only problem is that it will take more time than just to fly, but it is surely an adventure. What i would do is to take this pass until Fukuoka, stay there one night and than get the JR Railpass and hop on the Shinkansen to Kyoto.

Cheap airlines around Asia i know of: China airlines, China southern airlines, Thai airways.
I had planned a trip from Seoul to Bejing with China southern and a flight from Hongkong to Taipe with Thai and prices were reasonable!
oh, and i also think there is a ferry from Shanghai to Incheon.