Regarding cell phones, I believe your LG will work since Verizon uses the 3G network like they do in Japan. Softbank and Verizon also have a roaming agreement - as do most of the major providers in N. America.
However, I wouldn't recommend it for long term use because you'd probably be paying around $3.00/minute for local calls. I brought my iPhone 3G to Japan last year and used it three times. Twice to make a short local call and once to use my GPS since I was lost. That cost me around $10.00 in total.
Just go to a Softbank outlet and grab a prepaid phone. Here are the current rates :
Prepaid Service | SoftBank