Trains usually stop running around midnight-1:30 am or so, the shorter the trip, the later the train runs...if you're talking about a good long train ride, plan ahead so you don't get stuck at some random station. I'm not sure how much it would be from Nagoya to Aichi, but
a quick search found this
If you'll be traveling a lot in a short time, but want to take the scenic route, aka no shinkansen, consider seisshun juuhachi kippu. It was 12000 yen when I was there, and you get 5-24 hour all you can ride tickets. The catch is you have to use them up by within 2 weeks of using the first.
As far as food money, it all depends on how you want to live. If you're like me, you go out a lot and like trying random stuff. The times when I wasn't doing homestay, I'd spend a couple of bucks for breakfast, maybe some bread or onigiri and a drink, like 6 bucks on a lunch deal somewhere, then 8-12 on dinner. Of course, you can always cut back by eating convenience store food or fast food. I found the best way to save money and still go someplace nice is to pick up those free coupon books they distribute at some major train stations. I can't remember what they're called, but they're colored by season, and have TONS of deals on stuff you might otherwise spend a fortune on, like tabehoudai yakiniku, shabu shabu, sukiyaki and the like.