I’m a (strict) vegetarian and have travelled to Japan for work before.
Maintaining a strict veggie diet there is difficult. Not impossible, but difficult. I’ve spent time both in Tokyo and in smaller cities in Japan and the former has dedicated veggie restaurants, but they’re harder to find in the latter. Rurally it’s probably even harder. When you go to eat at traditional restaurants it’s very hard to explain the term ‘no fish’ to the kitchen. I had Japanese colleagues with me every time we went out and they spoke to the chef or waiter in Japanese, but my food still came covered in fish flakes, the soups were never free of fish stock, etc. It’s simply their way of cooking and it’s very hard to ask for something that’s not easily understood as a concept.
There was always some variety of rice bowl with egg and veggies, tempura, and 7-up’s and other corner shops and supermarkets have veggie foods. There are a few types of onigiri and sticky bean sticks etc, delicious. But in general I had to lay the strictness off and accept that some food would contain fish - if I wanted to eat at local places and not be limited to the same dish every day for 3 weeks. I could’ve gone to McDonald’s and Italian restaurants, but I wasn’t in Japan to eat burgers and pizza.
I had coeliac colleagues and it was hard for them but also not impossible. It depends a bit how strict you need to be with cross-contamination. One particular colleague was fine as long as no gluten was ingested so stuck to dishes without wheat noodles, another had prepared a list of safe restaurants from online guides and blogs, because she could only eat from a freely gluten-free kitchen.
Korea I believe is still similar (haven’t been recently) but with beef/chicken instead of fish everywhere. If you’re in Seoul or other major cities, there won’t be many issues. You just can’t be that spontaneous and walk into any traditional restaurant you like the look of.
All that said, I wouldn’t let the diet restrictions put you off what is an incredible part of the world. Do your research thoroughly, decide how traditional you want to eat and how flexible you want or can afford to be with your diets, and GO. There will be food, it might be a bit complex finding it but it’s worth it. The food is delicious and I’ve never come back from Asia hungry.
Cost - it can be expensive but there are ways of minimising this. Look at flying indirect instead of direct. Summer will be hot and humid - my favourite time is autumn - but it depends on how flexible you can be with school holidays. Look at flying via Amsterdam / Paris / Frankfurt / Helsinki or the Middle East. Look at multi-city fares and play around on Google flights or skyscanner a bit to see if you should be a return in and out of different countries plus Japan-Korea single fares, or three tickets linked together…
I can’t comment on internal transport costs as that’s always been covered for me, but public transportation is highly efficient.
Cultural norms I’d read on before you go but there isn’t really anything you need to change in your behaviour. Make sure you don’t eat or snack on the street, that’s the main one for me. It’s a very different culture and you might be surprised by some of it but as long as you’re respectful and attentive to people and local customs, just as you’d be in Europe, you’ll be fine.
I haven’t been to Japan since 2019 and I miss it so much, it’s wonderful. I hope you can take this trip and enjoy it as much as I always do. I could go there every year, there’s so much to see and do!