Hopefully you've done diags and had the system refilled first. Various components such as seals letting gas out, compressor clutch and valve can stop the air con starting up, and some of those can be replaced to fix.
If it's definitely the compressor itself, then ensure the whole system is flushed out thoroughly because according to research I did on Club RSX (such as
http://forums.clubrsx.com/showthread.php?t=823176 &
http://forums.clubrsx.com/showthread.php?t=733047) among other sites stated when the compressor fails it spits metal shavings into the system and if these are left it'll kill the next compressor you replace it with.
Mine went some years back and Honda wanted £1200 I think just for comp, labour and regas, don't even think there was a flush. So I imported a new comp instead, and took it to a commercial van/lorry air-con specialist - and saved half that cost in total. Made sure he knew what he was doing, he assured me his flush should get all these fragments. Also said to replace the moisture drier unit. Evaporator/condensor they said wasn't worth replacing as the flush should get to that too and behind the dash so would be a major pain to get to. Expansion valve not too important either if I recall.
Just as well I didn't give it a go considering the above and the tight placement of the comp which I remember them saying was the hardest bit of the job to get out without taking out the radiator and fans too - took them a week if I remember (fitting it around their other bigger jobs I'm sure). Also had them replace the aux belt too - think they would have replaced any o-rings needed.
Receiver/drier unit - £20 from
http://www.autoairconparts.co.uk/car-air-conditioning-receiver-drier-honda-civic-hatchback-01-05-2-0-type-r-petrol-manual-automatic-sep-2001-to-sep-2005.html. AirCon still working ok now.
List from the RSX links above:
Condenser/evaporator unit (not worth it)
Accumulator/drier/reciver unit
Compressor (should have the pulley which contains the clutch)
Expansion valve (I didn't bother with)
O rings
PAG oil
R134A gas
Long post but worth relaying the importance of my experience.