The question is legal equality, which seems possible and not depend on color of your hair.
Of course some Japanese will treat you differently. They will also otherize and treat differently even their fellow ethnic Japanese of other social classes. But I think this is not really different from British, Russian or many other cultures (maybe except US or Canada which are more melting pot like). It's just harder to notice from inside the culture.
I'm really not sure about that. I'm a white man with red hair, I'm pretty sure I'd always be "otherized" in Japan, as would most westerners.