Wow that can get complex.
If you are seriously thinking of it check with a reputable handler.
Japan Tokyo Narita (RJAA) is the most common stop. I believe paperwork needs to be in at least a few weeks before going. A wild guess at handling/ATC fees would be 10-15K, I have no idea about fuel. Time slots /curfews are critical. The Japanese have thier $hit in line and dont understand change most of the time. The actual flying to and from is completly painless and simple. The only headache is taxiing when Narita is busy. It can be downright comical. For hotels, the Grand Hyatt downtown is very nice. It can be an hour+ drive downtown.
China is a very common stop these days. US corp aviation is heavily present. Today we run into fellow pilots in Shanghai like we used to in Palm Beach! Again paperwork is critcal and fees are high. The Chinese also dont understand change and will just shut you down if they feel bothered. Handling is great, services are present and very adequate. Routing can be a bit strange at times, be prepared to be kept at very low altitudes for no apparent reason (read plan extra fuel) Security is great and hotels in the major cities (ZBAA,ZSPD,VHHH) are among the best in the world. It is a fast-paced, growing, driven, business minded society now. It is not you dads "red china" anymore.
If you have the ability, search the Intl database section of the NBAA website. It has great info on specific airports written directly from pilots.
Sorry I dont have more info on costs. Asia is one of the most expensive areas of the world to operate in.
All I can say is.... "if ya gotta ask..."