Re: Training at Hombu
No specific recommendations for places to stay, but you should be able to find something reasonable for $50 per night. Accommodation is cheap in Japan.
I second what Guillaume said about trying downstairs first and then seeing if you would like to try one of the regular classes. At 5th-kyu I think the upstairs ones might not suit you, but I haven't seen you train, so it is hard to say.
As for getting out of the city, I agree with NagaBaba. Stay in Tokyo for a few days to train at Honbu and get your geek on in Akiba, but otherwise it is a horrible place. (I lived there for a bit over a year, and just moved away last week). The castle in Himeji is beautiful, but I believe it is being renovated at the moment. Kyoto is beautiful of course, and you might also like to check out Hiroshima and Miyajima. Otherwise, it depends on how much time you have, but I thoroughly recommend getting out in the country a bit. I've heard that Shirakawago in Gifu prefecture is nice. There is a similar place called Ouchijuku in Fukushima prefecture and I thoroughly enjoyed going there. Also, make sure you check out the hot springs. You might have to get out in the countryside a bit to find the nice ones, but it is definitely worth it. If you are going to Nikko, there are some nice ones in that area. Save a bit of money when you stay in the city and use what you save one a nice Onsen-Ryokan for a night. The food is great, you get to sleep on a futon, and you get to check out the awesome baths.