I've been to Japan! It's one of my happiest memories.
Another cultural thing to remember, please don't stick your chopsticks straight up in your rice. That's a super offensive thing to them, since traditional funeral practices usually have ceremonies where they stick incense in rice. Mimicking this with chopsticks is considered disrespectful to the dead.
Have you considered Geru? I fell asleep in the car on the way up there so I can't remember how far it was from Tokyo, but it's a place known for its hot springs and has some pretty traditional hotels. Are you willing to try a public bath house? I was pretty nervous about it at first but the longer I was there the more I was able to relax. It's kind of like a spa, if you're into that kind of thing, so it's definitely worth checking out. But don't bother if you have any tattoos. Those are considered unclean, so you will not be allowed in if you do.
While you're in Tokyo, be sure to check out Harajuku too! It's a bit tourist-y, but they do have quite a few interesting shops and souvenirs if you're wanting to take anything home. I went in 2007 when goth and emo were kind of a thing, so it was pretty interesting to see the the Japanese take on fashion too.
As far as karaoke goes, it's everywhere. Some of them are kind of dingy, but there are really nice ones too. If you're under 18 (which I was at the time, though my birthday was really close) it's illegal for you to stay in the karaoke buildings after 10:00, but any time you get the chance to do it it's pretty fun.
Osaka was as nice city, but it felt kind of... I don't think dangerous is the word, but it felt less safe than the other places we went to. I should probably take this time to mention that for your own personal safety, stay as far away as you can from a person with four fingers instead of five. Japanese organized crime members will cut off one of their own fingers in loyalty, so if you see someone like that be on your best guard. (We actually were approached, but my teacher started speaking German to them and they went away.) Another similar place that felt much safer to me is Kyoto, the third largest city in Japan. Maybe you should try to hit at least Tokyo, Osaka, and Kyoto while you're there, since they are the 3 largest cities and they have quite a few things for you to do.
If you want to see Nara, then I say yes. The little deer are just too precious and they come up to you and are so friendly. (Though we won't speak of the teacher who came with us and kept getting bitten in the butt. One of them really wanted her cookie.) For the most part, they are friendly anyway!
Make sure you see as many shrines and temples as you want, but don't go every day like we did. Make sure you are doing new things to keep yourself interested. It got to the point where we quit looking forward to them, and I hate that. Going to another country should always be interesting... the Nara shrine was a win as was a temple in... I think it was Nagoya? I don't remember. I will go through my old pictures for you if you want me to.
The food is really yummy too! Go in with the mindset that you will never be able to try something again, so you might as well while you have the chance. "Street food" (think of them as hot dog vendors) are more prevalent than in the US, so be sure to try lots of things. Okonomiyaki is good- they are eggy pancakes with mayo and whatever you want to put on it. Okonomiyaki actually means "grilled whatever," so don't be afraid to try weird combinations that sound good! Dango is good too. They're colored doughy balls. I don't remember what flavor white was, but pink was sakura flavored (cherry blossom) and green was green tea flavored. The green ones are to die for, by the way. But by far my favorite thing was takoyaki, which are little balls of grilled dough that have just a tiny piece of grilled octopus inside. Some of them are served with a barbeque sauce too. I know it doesn't sound good, but it totally is! Octopus in things is pretty good, and kind of bacon-y, but just plain octopus is kind of like chewing on a rubber band.
(The family I stayed with really wanted me to try octopus since it's my favorite animal and I made the girls little yarn octopus dolls.)
I know this was kind of all over the place, but I love Japan. Best of luck on your trip and please show us pictures!