Well, I haven't been 18 or 19 in 18 or 19 years so I don't know if my suggestions would be welcomed or not. But having been to many of the great cities in the world and seeing today's youth, perhaps a trip to Shinjuku Gyoen is in order. This is a large park near Shinjuku Station. It costs 100 Yen to get in and is pet free. Just go with some food and drink and relax out on the grass. You'll find plenty of youth out there as well as families. It would perhaps be similar to Hyde Park. I've been to Japan some 15 or so times and the park is still one of my favorite places to relax.
More fun could be had at a karaoke club. The rooms are small and will accomodate you and your friends. Loads (thousands) of English songs available to choose from.
Not sure what the Japan baseball or football schedule is like for early Spring, but taking in one of those sporting events is fun, especially baseball (being English, it may not be as interesting as it is for Americans). J-League football is always improving. Tickets aren't too expensive.
In Tokyo, get subway day passes to really save a bundle on transportation.
Most of the top attractions in Tokyo are free so you don't have to worry there.
For food, look for "Otaru Zushi" for some dirt cheap sushi (105 Yen/2 pieces). I went to the one in Tsukuba and was pleasantly surprised. While in the fish mood, get to Tsukiji fish market early. Oddly it seems to be the most memorable thing for many travelers. If you want to have some great tasting food in a unique restaurant, I suggest the Maisui restaurant near Shimbashi Station - (use google and find maisui - this forum won't let me post the URL since I don't have enough posts). Avoid fast food places (McD's, Mos Burger, etc) as they are expensive for what you get. Small noodle shops provide more food for the yen.
To stretch your money farther, remember to exchange money at the airport in Japan. Not sure if it's the same in England as here in the US, but the rates we get at the US airports is lousy. When I take traveler's checks, I get close to 99% exchange at Narita. Cash is 97-98%. Once you get to hotels, you still get better exchange rates for travelers checks but usually the rates are a few percent lower.