While we offer a tool to calculate an estimate budget, this page will give you pointers on how to go as cheap as possible.
Estimate Budget Tool
Air Travel - While it would be best to choose a season you like, prices can get low on the low-season. Also be always on the lookout, as air companies often offer temporary discounts. Don't be afraid of long trips, you are going to stay in Japan, so what is a day worth of travel to get there?
Days to stay - It is an often chosen approach to match the days you stay in Japan with one, two or three weeks to match the JR-Pass validity. However, a good way is to stay in Tokyo in the first or last days outside the pass, since it won't be much useful for metro/local trains.
Stays - You can go any price on this, and other than the absolutely super cheap, most stays are clean, safe and reliable. However, be aware that AirBnB is not regulated in Japan, and since it is illegal to rent to foreigners, Hosts can get in trouble if they are caught, so it is not uncommon for AirBnB's to cancel without notice. If you are going with AirBnB, try to have a backup plan. Also, to get cheaper, try to stay away from the city centers, of if possible, on nearby cities. For instance, Osaka is cheaper than Kyoto (and only 30 minutes away), Nara and Kobe are even cheaper (only 45min from Kyoto). The same holds true to most cities: nearby cities are often cheaper and might be close enough to make it work.
Daily Expenses - Japan has good quality food for as low as ¥300 (seriously), so don't worry that you will starve. Needless to say, those meals are not really nutritive or fulfilling, so you can plan around. A good mid-term is ¥1.000 a day for two meals. People often question this number, but its just a matter of wanting to eat too fancy. Get to a 7-11 or any combini and get a takeout meal and you will see this is actually on the spot.
Plan for entrances - Most places require a small fee. They can be as low as ¥100 for gardens and parks, but can go up to ¥2.000 for modern or fancy observatories or attractions.
Remember to carry some cash for local trains not covered by whatever pass you are carrying, if you have one.
And make a travel insurance, even the most basic. And a word of caution: those travel insurances that are "included" in credit cards are terrible, don't cover everything and will require "proof" for everything (a family friend had a stroke in a trip and require specialized medical care and couldn't travel back, the credit card insurance had the nerve to ask "proof" it was not a pre-existing condition ... a stroke? And they did not cover repatriation: he needed air-ambulance place, and the insurance didn't cover, so he had to stay away for the whole treatment, and not a surprise, the travel insurance didn't cover the trip of a family member to stay with him.
You can plan all you want, and cut costs and make lots of savings, but remember not to cut costs on anything that might get you into trouble, or escalate to something expensive.
In time, it is perfectly ok not to plan. Japan is usually a little more planned because of all the cities, places and offerings you can cover in a trip, so people looking to get the most of their time will build intricate itineraries and research deals, but if you just want the fun of find hidden gems and plan as you go, you can do that. Just check at least if a Pass can give you savings, and for any better stay, the advice for early book stands.