i've stayed in various internet cafes overnight on a few occasions when i lost track of time and missed the last train home.
while overnight stays are not prohibited, and im sure plenty of foreigners have traveled this way...i would not recommend it for more than a single night at a time.
most cafes that i've been to are cramped and tight. add luggage and other people, and its going to get even tighter and more cramped.
also, unless you can speak a decent amount of japanese, be prepared for some awkwardness at the front desk. remember these places aren't necessarily designed with foreigners in mind, as opposed to a hotel with english-speaking staff.
i guess it is certainly possible, but i agree, save up your money a little more and use the internet cafes as a last resort (no pun intended).
why not try hostels, or the capsule hotels?? they will be cheaper, and are designed for overnight stays. i've even heard of people staying at Love Hotels (minus the Love) overnight because some are cheaper than business hotels..
hope this helps.
