Haven't been in about a decade or so so my advice is fuzzy and out of date, but:
-JR Pass was worth it for not just the savings but also the convenience of it, but if they jacked the price up hard to say. Will say it was super convenient to not worry about getting tickets though and just being able to hop on a train especially when you're with an indecisive group that wants to zig zag around Tokyo like maniacs. There are some lines that aren't JR though so it's not usable everywhere, I think I remember being able to get around Tokyo with no trouble using it though.
-I traveled with just a small over shoulder bag and would recommend getting as small as you can manage for luggage. From watching my friends struggle I can say that dragging a giant suitcase around the train stations can be a bit of a nightmare.
-Kyoto is one of my favorite cities there, but 4 days might be a bit much. It's a great place to visit between Tokyo and Osaka though as a bit of a breather for the much more chill atmosphere.
-I was a broke college student so stayed at mostly hostels so not sure on hotel recommendations. If you're looking to meet people while you're their I can't recommend hostels enough though, most that I stayed at were pretty decent as far as hostels go.
The one I stayed at in Kyoto was super nice and above a pottery shop/bar which can't ask for a nicer combo! Can try to hunt down the name if you're interested.
-Osaka is great for the night life and some shopping, love Dotonbori. Different vibe than Tokyo for sure
-Nagasaki is my favorite city, but not much to
do
there aside from relax so it's hard to recommend unless you really want to just chill for a few days. Has some nice onsens (public bath/sauna type of deal) with amazing views, some looking over the ocean and one that overlooks the city but you can find similarly great onsens in other cities too. Also some amazing views to see without having to be naked in an onsen, can't remember the name of it but you can take a cable car up above the city and the view is amazing at night seeing the city right along the ocean. Pretty nice restaurant up there too.
-I remember doing a bit of hiking and renting bikes in Nagasaki but would have to do some digging to try to remember where that was exactly.
-Baseball games are an experience I 100% recommend even if you aren't into baseball, the atmosphere and crowd energy with chants is something totally different than an American game
Have a friend over there right now sending me endless snaps of their trip so might finally cave and head back next year.