For all your Japanese-citizenship related questions, study this blog: http://www.turning-japanese.info/
No, Japan does not permit dual citizenship. (It's a bit more complicated than that, and enforcement is lax for citizens by birth, but "no" is probably the answer for you.)
Can you get citizenship through a grandparent? Well, you can get citizenship through a parent, who in turn can get it from their parent. Otherwise you'll have to go reside there for five years (less if you're married to a Japanese) first. There are additional requirements--again, go study that website.