The cheapest way for a foreigner to live in Tokyo is probably to rent one of those guest houses or shared rooms from a company like
Sakura House. The prices on those goes down below 40,000 yen a month for the very cheapest ones, but budgeting 50,000-75,000 yen a month for renting a cheap place might be better if you can afford it. You can take a look at the Sakura House website for their prices.