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Best Time Of Year To Travel To Japan?

I think 5k is quite high for 2weeks, but as mentioned it depends a lot on your lodging etc.
Just book your accommodation early. Current business strategy of many japanese hotels are cheap bulk sales to Chinese tourist operators and ripping of everyone else.
2010/11 was a paradise to travel Japan, not so much anymore. If you plan to visit during sakura / autumn season or golden/silver week (I think there isn't one next year) it's especially important to book early (read month in advance).

Nevertheless I think that you can easily get along with 1000€/week, less if you travel little or put emphasis on cost efficient accommodation, but obviously you can easily spend as much in a day.
 
Oh, 5k WAS high, but the first two times I came the yen rate was 70 to the dollar on a good day, which made everything cost a ridiculous amount. I think my flight was around 2k plus my economy plus upgrade, so the actual cost of being in Japan was lower, just total cost of trip was a bit high. But yeah, my trips were expensive, but they were all worth it. My 2014 trip for 18 days started in Tokyo for a day and a half (Sat night-Mon morning), then Mon-Fri in Hiroshima, Fri-Thurs in Kyoto, then Thurs-Tues back in Tokyo. Definitely got around the country that trip!
 
^^ If I remember correctly, things start to cool down as early as mid-september, but it's usually closer to the end of the month before bearable temperature comes in. I seriously wish I could just fast-forward the calendar right now by a month or so. I get to the station at 530 in the morning and I'm dripping with sweat