I really wouldn't worry too much about high humidity levels if you're going in March (heh). In fact, March is a great time to go as the spring thaw should be in full effect (not like Tokyo ever gets that cold anyways).
Re: drinking the water, I don't think that really applies to Japan (at least from my experience). I can't think of even a single instance where someone fell ill as a result (to my knowledge). Some people have been known to have difficulty with their bodies adjusting to the time zone, diet, and differences in temperature when staying extend periods of time but since you're only going for a week you should be fine.
As to where to go and what to see in and around Tokyo, well that really depends on what your itinerary is during your weeklong stay (shopping, sightseeing, nightlife, etc.). There are certainly plenty of things to do and see in Tokyo but if you're feeling a bit more adventurous, you could certainly venture a bit farther to check out places like
Kyoto. etc. Fear not, for the transit system is very easy to use and not difficult to navigate at all and should you ever have a question, the local citizenry will always be willing to help you find your way.
My advice to you is to just take it all in stride once you get there and do what comes naturally without feeling you need to adhere to any strict plans. You'll have more fun this way and be able to relax while enjoying yourself without worrying or stressing over "must do this asap, must be here at such-and-such time, etc." Oh, and be sure to eat to your heart's content while you're there and try everything at least once.