I'd almost definitely get a JR rail pass - it'll pay for itself with one trip. You have to get that BEFORE you arrive in the country. And then
it's a bit of a pain to go to get them validated at the start of the trip. Then all you have to do is to remember not to lose the pass!
There is an alternative for cheap travel on the trains called a 'Seishun Jyuuhachi Kippu'
or 'Young Persons-18 Ticket' as a loose translation. It's only available during university breaks, so prob not ideal
for you, but gives as I recall 5+ days unlimited travel on 'futsuu' (non-express, not Shinkansen/bullet) trains over a two
or three week period.
Japan is expensive, there's no way around that. Particularly accommodation (look at 6000-8000 yen a night).
Food is actually reasonable - no reason to spend more than 1000 yen on a decent meal. And on a budget you
can eat for 300-400 yen. Local travel within cities is cheap, it's just the long distance stuff on the shinkansen that's
expensive and sometimes comparable to flying.
Suggested places to visit:
For Cities/Shopping/Nighlife - Tokyo (Shinjuku, Harajuku, Roppongi, Ueno Park, Asakusa, Akihabara, Tsukuji),
Osaka (Ebisu, Hep-5, Umeda, Namba, Denden Town)
For Temples/Very Japanese Things - Kyoto, Kamakura (near Yokohama), Nikko (north of Tokyo), Nara (near Kyoto), Miyajima/Iwakuni (near Hiroshima)
For Remembrance - Hiroshima (Peace Park, Genbaku Dome and Museum)
For Food - Anywhere! But Nagoya (Miso katsu, also Unagi-don an eel+rice dish), Hiroshima (okonomiyaki), Fukuoka (raamen) and Sapporo (raamen and genghis khan bbq) have some excellent variations. Osaka has takoyaki!
For Mountains and Onsen - Fuji (either Hakone or Kawaguchiko), Nikko, around Matsumoto/Nagano, Takayama
For Castles - Himeji, Matsuyama, Nagoya, Osaka (in that order)
For Snowstuff - Nagano-ken (e.g. Hakuba), Niseko (awesome!), Furano, Zao (famous for its ice trees ('jyuuhyou' i think) - but you get that anywhere up to the north of the main island, but strangely not in Hokkaido.)
For Chinatown - Kobe, Yokohama
For Funparks - Disneyland Tokyo, Universal Studios Osaka, and I can't remember the name but near Nagoya with what was the worlds highest rollercoaster (may still be)
For Beaches - Okinawa
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