I'd also add that Love Hotels can be very difficult to get into on a Saturday night (especially the reasonably priced ones). And once you're in, you're usually in. You'd need a pretty good grasp of Japanese to be able to convince the reception staff to let you back into the room if you decide to pop out.
Perhaps it's possible to go and come out of a room as you please, but finding somewhere on a Weekend, that's affordable, and that lets you return to the room would be more than I would be capable of doing... and so I wouldn't advise this for a visitor.

Cheap places to spend the night include all night Sento (public bath houses - Green Plaza in Shinjuku and LaQua near Korakuen are examples I can think of from the top of my head). There are some all night cinemas in Shinjuku which seem very popular as places to crash out (as we were staying locally I went to one at midnight with my husband. We were the only people laughing, everyone around us was snoring... I think it costs 2,000yen and you can stay there all night. Alas, it'll probably be the same film on a loop.)

I'd say go out all night. Shinjuku might be easier on your pocket than Roppongi. Plenty of places to eat and drink. But check out something like http://metropolis.co.jp/ and find a club you'll enjoy and go there.
(Once you include the price of a few drinks, the clubs cost the same as they do in London - the entrance charges might seem steep but they include a beverage tokens. If you like cocktails you should get your money's worth.)

Get one of the first trains back to NRT and hire one of the day rooms there for a few hours and get some cheap sleep and a shower.
Terminal 1: http://www.narita-airport.jp/en/guid...st/svc_37.html
Terminal 2: http://www.narita-airport.jp/en/guid...svc_37_t2.html

Depending on when you're arriving, you might want to grab a couple of hours sleep before you hit the town. The rooms are available to arriving and departing passengers.

Have a wonderful time!