First, let's assume that you do not have millions of dollars at your disposal, and you don't want to stay for a full year. You just want to stay longer than a week. What are your options?
1. Getting in.
Upon arrival in Japan you'll pass through immigration. Tell them you're here on holiday. So long as you don't have a criminal past, you'll get your passport stamped with a tourist visa, which is good for 90 days.
Please note: don't try to overstay your visa. Japanese immigration takes this stuff very seriously, and you will not be happy when they find out. They may kick you out of the country and ban you from returning, or they may throw you in visa jail for a while.
2. Where to stay
Assuming three months in a hotel isn't in your budget, you could stay in a hostel. Hostels are cheap, and easy to arrange through sites like hostelworld.com. Plus, you've got an ever-changing international cast of characters to make friends and hang out with.
A bit more expensive, but also a bit nicer, is the option of a share house. Share houses are usually smaller buildings in residential neighborhoods where you can rent a private room or a shared room. Bathrooms, shower rooms, and kitchens are for common use. There are loads of share houses here. Want me to Google that for you?
People who live in share houses are mostly international students, international English teachers, or young Japanese professionals who are looking for relatively inexpensive living accommodations. In other words, this is not like living in Animal House. This is not a place for you if you want to go get wasted every night, stumble in a 2am and blast Foreigner's Greatest Hits.
Finally, there are companies that specialize in short-term leases for foreigners. Right now, I rent from Fontana. I like Fontana because they were really helpful and kept the whole process was simple and stress-free. Plus, Fontana does 3-month leases, which is nice. The accommodations are by no means luxurious, but for about $750 USD/mo. (which I can afford because I have a job), I have a furnished apartment in central Tokyo. It's tiny, it's minimal, it's quiet, and it's mine.
You gotta keep it cheap, right? You're in luck. Lawson 100 is sort of like a convenience store meets grocery store meets dollar store. Pretty much everything in the store costs ¥100.
Now, before get all yicked out about dining at the dollar store, remember this: the Japanese don't go and get a week's worth of groceries on Sunday morning. They buy food every day. Freshness is everything here, so you are not purchasing day-old bread and nearly-moldy meat. It ain't Whole Foods organic hippie granola, but it is fresh and tasty. ALL of my groceries come from Lawson 100, and I spend maybe $10/day on home-cooked meals.
4. Getting a job
Many countries have working visa deals with Japan. The U.S. is not one of them. Unless you're fluent in Japanese, the only real option you have is working as an English teacher, and you're not getting that job if you're only staying a few months. So save up some money.
I think that if you were really careful, walked a lot instead of taking the train a lot, cooked in instead of eating out, and didn't spend lots on crappy souvenirs, you could get by on $1000/month (not including rent). That means you're not clubbing much. If instead you like doing lots of long walks, exploring, hanging out, writing, and reading, then you could make it happen. It would take discipline, but it could be done.