I've no idea where you're staying but from past experience October in Cyprus can be quite warm, unless you venture up to the mountains. Frozen hail-stones & 4 degrees in Troodos (when compared with 20 degrees in Pafos) was a surprise, especially 'cos we weren't properly dressed....
Ayia Napa is OK in the off-season, Cape Greko & Protaras probably worth a visit too.
If you are going to rent a car, (I always do if only to get to / from our digs.) you'll find Cyprus is a fun place for drivers, a really useful motorway network, entertaining main roads & some twiddly routes through the mountains. It's a country of many contrasts, from posh shopping centres, supermarkets & upscale beach resorts to derelict villages (remnants of the Turkish / Greek conflicts) in the hills.
The EU has funded much improved public transport too (no trains though...), cheap local buses + long distance coaches too.
If you do want to visit the Turkish occupied zone, it's best done in Nicosia, just walk north on Ledra Street (pedestrian only road) there are Customs Posts either side of the Nato no-mans land. (Bring passport, of course.) There are other border crossings that would require a car to reach them, but rental cars can't visit the Turkish zone, 'cos they wouldn't be insured.
I've visited Cyprus 6 times, so I've learned a little about the island & since I'm not a beach person, I'd really recommend some exploring.