Tokyo Metropolitan government building in Shinjuku - Free trip to the 52nd floor, incredible views... the building is like something out of bladerunner aswell. Shinjuku - busy shopping district and some awesome ramen shops there Akihabara - Electronics district - Anyone who likes gadgets and electrical goods will be in hog heaven here Harajuku/Yoyogi Park - Loads of young japanese people here, loads of trendy shops and people dressed up in fantastic costumes.... a must see. One place you should see but dont spend much time in is Roppongi - the "bigg market" of tokyo - full of annoying expats and australians... Be prepared to pay 6-7 quid a drink there, so dont bother with that much. Best places to drink are "nomihodai" - 15 quid to drink all you want for 2 hours, although i think these are a summer thing only. Apart from that, find a good Izakaya (japanese pub) and you get little snacks and food aswell as big mugs of beer in frosted glasses brought to your table,... figure 2-3 quid a drink there - best places to go imho. I'd recommend either the "Rough Guide" or of course the "Lonely Planet" books for Tokyo - both great insight into the city.
One of my pals went there on a works placement for a month then got offered the job and stayed, he loves the place.