It's Really Safe
Japan is among the safest countries in the world, and while it doesn't mean you should throw caution to the wind, you definitely don't have much to fear. (A relief when you're a woman by yourself in a foreign country.)
In fact, you are probably much safer there than in any major city in the US, since Japan has the second-lowest homicide rate in the world. It's common to see small children taking the metro to school and running errands alone. I actually felt much safer there than I do at home in LA!