Roppongi Hills is one of the largest new developents in Tokyo, with the massive Mori Building at the centre. JPY1,500 buys you entry to the Tokyo City View, offering spectacular views from all directions (depending on the air pollution. You'll get the best views in February). On the same floor you'll also find the Mori Urban Institute for The Future, which has incredible fully-detailed scale models of both Manhattan and central Tokyo, and is likely to bring out the Godzilla "must destroy this thing!" instinct in you.
Directly connected to Roppongi subway station (Hibiya and Oedo lines).
www.tokyocityview.com
That money doesn't just buy the view. The art gallery is really excellent, with some of the best contemporary art I've seen in Tokyo.