Japan is my favourite destination. Leave any prejudices at home and you will love it! I am travelling so on the tablet and difficult to type but My reco’s would be:
Look into JR pass (Japan Rail), you can only buy it outside of Japan as a tourist. It is not cheap but it gives you unlimited travel on fast train network. In that case it makes absolutely no difference where you are flying it is only a couple of hours on the train wherever you go really. It also gives you access to wherever you want.
Second, if you do a bit of research it is not an expensive place at all. You can get a katsudon (rice and breaded pork) or a bowl of ramen / curry for cheaper than the uk. Of course, it is not all cheap and yes you can get adventurous and eat fugu (puffer fish) sushi for considerably more. Try to immerse yourself and not go to McDonalds
make an ar$e of yourself in a karaoke bar or play pachinko in ear-busting hall and you’ll like it a lot more.
Another phrase you’ll learn quickly enough as a westerner is “Nomihodai” means “drink all you want” in an lzakaya (a Japanese version of the pub). Great fun.
Tokyo also used to have free guides (people who show you local sites and take you around. You buy them lunch and speak English to them it is great. If you are into more up-market side of things, i have a friend (gaijin - foreigner, from Exter, as it happens) in Tokyo who does high-end personalised tours, but he is defo not cheap. PM me if you want more info.
Hope that helps. My advice is add as much time as you can to your trip to go around, for me it was a really great trip. Enjoy!