Not sure what your age is but we met some guys 2 years ago from Ireland (in their 30's) who HATED Ayia Napa - too noisy, brash etc and that was in October!
You won't find the "real" Cyprus there but as already advised you can get a hire car and head for the hills to the mountain villages. It IS quiet in the villages but Ayia Napa in season is the complete contrast. Sept is still pretty hot so you will get end of season drunks doing the usual sunbathe/get burnt all day and get drunk all night brigade.
My advice, get out and see a bit of Cyprus as well as exploring the area around Ayia Napa - there is some good stuff to see/things to do.
Check out the list of things to do on the website below:
http://www.cyprus-travel-secrets.com/10 ... to-do.html