I have to go into a trip while being slightly ill - doctors say I have to drink a lot. Can I buy water at a duty-free shop, bring it onboard and drink there?
It depends on the origin and destination airport
For most flights, liquids purchased after the security check can be taken on-board the plane. Not all though...
For an example, at the moment (2011), for flights to Australia, no liquids (beyond the 150ml limit) may be taken onto the plane with you - there's an additional security + liquids check at boarding. Duty Free may be able to be placed into the hold (depends on the airport's setup), but you can't take liquids (even a water bottle you've just bought) on with you.
So, check with the rules for your specific destination and origin before you go!
One other tip though is that you are normally allowed to take empty bottles through security checks, it's only the contents that's an issue. So, put a few empty water bottles in your bag, and once you clear security find a water fountain and fill them up. As long as you're not flying to somewhere like Australia, you'll then have a decent amount of water with you for the flight. You may also then be able to ask the crew to fill one up for you mid flight, to avoid only getting water a cup at a time
Aircrew aren't worried about your carrying on "liquids", they're worried about your drinking alcohol. If someone is "getting high off his own supply" as Tony Montana used to say, they might have trouble cutting him off if he gets unruly.
Presuming your doctor just wants to make sure you have plenty of water, just bring an empty water bottle and explain your problem to the purser or one of the flight attendants and I'm sure she'll be glad to fill up the bottle for you from the aircraft's own fresh water supply. If you're at all concerned about it, get your doctor to write you a note on letterhead stationery.
Be aware, though, that in case of turbulence, you won't be allowed to use the lavatory. You may be very uncomfortable sitting there with a full bladder, waiting for clear air so you can make a run for it. Word to the wise.
In any case, duty-free doesn't sell water, since there's no duty due, but there will be shops and stands that sell bottled water. As other poster have pointed out, bringing the water aboard may be another question. Of course, few airports institute a second search once you're passed security -- the only one I know of: Suvarnabhumi (BKK) in Bangkok.