Well I have an exedy on my 92 gsr and we just installed the same kit on my friends 94ls. They are both awsome. He used a Type R flywheel, I would imagine that both the CTR and ITR flywheel would work for you civic. Otherwise Exedy and Fidanza make excellent applications. As far as cost the average I have found is between 300-375 which includes resurfacing of the flywheel(if not a new one, like a used type R but not needed if u get a fidanza for example) and labor.
However, the actual process of replacing the clutch is not that difficult, just time consuming. It took my friend and I collectively 7 hours or so spread over a weeks time period. We had to get a couple of flywheel bolts because his were ground down. Ive never really looked into an 01 civic engine bay to tell you whether itd be just as simple, but if it has as much or more room in the bay as the 3rd gen integras, you and a buddy could do it yourselves. there is an indepth article on team integra with pictures and step by step instructions. That is what we used. Just get a good breaker bar and a torque wrench, also some grease for the clutch and make sure you will have access to an auto parts store, or honda/acura dealer. good luck.