Question

I am planning a longer trip of about 8-9 weeks in japan. what should i expect on this trip and how much money should i approximately...

...carry with me and what do things typically cost. What is the best way to carry money? Cash, TCs?

Location

Country: Japan

Answers

It really depends on how much you want to "do" and travel and where you go. Japan is not a big country but it covers a long distance north to south. Climate, customs and prices vary by prefecture/area. I would estimate $10,000 at least for such a long stay. Hotels and food being your biggest costs.
Welcome to Japan.
8-9 weeks is a long period. Depends on how many days and until where you wish to travel. If you wish to know more interesting attraction of Japan, check out the official site

A simple calculation. Normal budget hotel/hostel will cost about 3500-5000 yen per day. A decent hotel might cost 10000 per day. So, your accommodation will go from 220500 - 630000 yen for 63 days.
Average dining cost is about 2000 yen for traveller. Dinning cost is 126000 yen for 63 days.
Travelling in Japan cost about 3000(short distance daily trip) - 10000(long distance).

So, what you need will be at least 346500 - 756000 yen ( about USD 4000 ~ 8300 ) + days * travelling.

Never bring that much of cash with you. TC will be good but you need to cash them at bank. Credit card is acceptable in major hotel and supermarket. Cirrus ATM can be located easily too.
You can save a LOT of money by purchasing a Japan Rail pass in your home country (it CANNOT be purchased here in Japan). It allows you to travel on most of the bullet trains which are the fastest and most convenient ways to see different parts of Japan. However, the rail pass must be used on consecutive days. I still think it's a great value, considering how much domestic travel is for non-tourists.

There are many capsule hotels, but you may also want to contact local government offices and see if they maintain a list of families who are interested in hosting short-term homestays for foreigners. Some Japanese people are excited to have any chance to interact with overseas visitors.

I also recommend Japan National Tourism Organization (JNTO)'s webpage for narrowing down your travel plans and figuring out how to get there. There are many printable pdfs that list accommodations, major attractions (including admission fees, hours, directions, etc.). Good luck!

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