1) What to prepare
・TOEFL（iBT>100）
・GRE (General and math. General exam is very demanding, so usually you need a year to prepare for it)
・Recommendation letters (they would ask for 3 of them. You may want to ask the professors who know you well for these letters)
・GPA (If you are aiming to go to top-tier schools like Harvard, MIT, Stanford, you need a GPA> 3.8)
・Resume (lab experience preferred)
・Letter of intent

After application, there will be an interview session if you are selected. Please seize the chance to actually go to their campus, talk with professors and current students there. If you have multiple interviews, you can even have a comparison.

2) Scholarships
You can search here. If you can have a scholarship from Japan, it would be a huge advantage. So please at least have a try. Many of them require Japanese nationality.
http://ryugaku-shogakukin.jasso.go.jp/scholarship_abroad/page?action=swfglsearchjasso

This is an organization of Japanese grad students who are studying in the US right now. Please see their website too. You can find a lot of useful information from this site.
https://www.facebook.com/gakuiryugaku/?fref=ts

3) If you want to apply for a grad school in the US, you need to begin preparation really early. For instance, a lot of people begin to prepare in their freshman or sophomore year. And you need to have a high GPA and nice scores for TOEFL and GRE. It would take a huge amount of time. Prepare as early as possible.