For someone who is a quarter Japanese and has a native grandmother in Japan...

Does that give me any options with Visas and extended staying? Sorry I don't know much about this area.. at all.

Looking at the MOFA website, you might be able to apply for the Family Stay or Long-term Resident visa. However other websites don't seem to have any information about the Family Stay visa. Try enquiring at the embassy about the Long-term resident one.