Hostels are pretty nice in Japan like everything there. Only thing worse compared to a hotel room is like shared bathroom/shower and just a small square. Otoh, you can book hotel rooms for like $60-80/night in most places.
You can definitely do 2.5 weeks on $2,500-$3,000. Like just check out expedia or any of those sites for hotels in Tokyo or whatever city you want to go to and you'll be able to figure it out. Main thing you want to figure out is if you want to travel around on bullet trains and check out different cities in which case you wanna get a rail pass. If you just wanna explore Tokyo, you don't need one.
My typical "show people Japan" tour I give when I travel with first-timers is:
Land in Tokyo
Tokyo wards - Shibuya/Akihabara/Harajuku/etc...
Tokyo outer area - Odaiba, then whatever wards left, Tokyo Tower/Skytree
Do Fuji/Hakone if you want
Or take a day trip to Nikko or some of the <1 hour cities
Head to Kyoto/Osaka
Do a day in Kyoto, see the temples, go to Arashiyama with the long bridge and bamboo forest
Do a day in Osaka, do Den Den Town, Gundam store, Namba
Head over to Nara for the deer, or go to Hiroshima/Miyajima
Stuff like that. I personally like to take a day where I take people out to a small rural town to explore because town living in Japan is totally different than cities and a cool experience. You could always get an international driver's license (just fill out a form at an AAA) and rent a car if you're brave. Just do not rent one in a big city cause that's crazy. If you do, take a train out of a city to a smaller town and rent one so you get mostly open roads, but the tolls are like $100 for an hour or two drive so maybe that's not really a great idea.
Ryokans are really cool, and Onsen Ryokans are even cooler. But they usually don't come cheap. If you want to splurge for a night, that's the place to splurge.
Japan has basically 1) Amazing food, 2) Gorgeous nature & history with hikes, temples, hot springs, and 3) Awesome videogame shopping, so just look around for stuff and make your own schedule. If you're in a city you don't need to plan ahead, you can just walk around and explore but it'd be good if you had wifi dongle thing and a phone so you can always google what's nearby.