Unfortunately, there isn’t an exact answer to this, as Every Situation Is Different. You will have to bring enough money to survive your first month. Enough to pay rent and utilities, buy groceries, and furnish your new apartment.
So, how much do you need? The General Information Handbook recommends ¥250,000 (about $2,500 USD), which should be more than enough. Depending on your rent, possible rental deposits, and the date of your first paycheck, this amount will vary considerably. Your contracting organization and your predecessor should be able to give you an estimate on rent and other costs, so be sure to ask them.
We have heard tales of new JETs surviving on only ¥50,000 or ¥100,000, but it’s better to be safe than sorry.
You should bring at least ¥30,000-¥50,000 in cash. Japan is a relatively safe country, and it is typical to carry around this much money. The rest can be brought in travelers’ checks or withdrawn via an ATM card from your home country bank. Travelers’ checks can be cashed in all major cities (including Kobe), but may be difficult to cash if you are in rural areas. You can withdraw money from your bank account in all postal ATMs, all ATMs in 7-11s, and some other ATMs.