I've ridden bikes all over japan, not touring style, but just ridden. You are not likely to get stopped by Japanese police while on your bike unless you're being stupid. But they do have 'meter maids' that go around and check for the registration stickers when bikes are parked. Just ask your friend where to get it registered. I bought a cheap bike from a store and it came registered. And make sure you park in designated areas, or you will get a ticket. It's a sticker and a pain to get off. I never paid them, but they didn't know who I was anyway.
I don't know much about the Kanto region either, but I did a mini tour (3 days) across Hiroshima on a used kids bike that I bought for $15 at a recycle (pawn/second hand) shop. I was originally going to set out on the main highway, which I think it illegal and some drunk old guy advised me to take the secondary highway instead. Much safer, and better scenery. Glad I took his advise. Kinda the equivalent of Route 66. Take the non-tolled highways in Japan and you should be able to travel efficiently from point a to b, with safety and great scenery. If you'll be travelling over the course of a few days, bring a tarp or tent. Depending on the season you are going, I would advise you to be aware of rainy season in Japan. A month or so of almost non stop rain.
and if you're bike is too big for the car, consider asking your friend to skip the car and take the train with you and help you carry your luggage. There is the Narita express (slightly more expensive than the regular train), which hits all the major way points to transfer and get you anywhere you need to go in the Tokyo region and beyond.
Tokubin and other delivery services are available just outside of baggage claim at Narita. Probably a 3000 yen charge for next day delivery. Offer to have your friend meet you at Narita by train - easier for everybody.
Just get on your bike and ride. Police will avoid you and nobody will care that your bike isn't registered. Make sure you lock it though, because borrowing bicycles is sorta not considered a big deal in Japan. Also lock it where there are other bikes, not just "barra barra". The borrowers will take the ones that aren't locked.
As far as routes go, stay off the highways and just follow your nose. If you can't read kanji, then maps are useless. My guess is that after a while you'll start using local trains like everyone else.
Very good point about making sure your bike is locked. Bike theft is very common in Japan, and the police don't know if a bike is stolen or not when they see someone on it, so it's not a very good chance of getting it back. and use a good lock. the generic lock they use in Japan is very easy to pick.