Both of you need to trot along to a Muhtar (together) with your I.D.
All car documents (just to be safe!) - I'm not sure about proof of address, so use your own judgement on that...
Muhtar will complete a form stating that person A wishes to sell the vehicle to person B, stamped and signed. Pay him a couple of Lira or whatever for the stamp.
One of you, preferably the seller! (to make sure that you don't get stung with the new driver's speeding fines ) needs to go to the Vehicle Registration office in Lefkosia - it's not too far from the 'Ministries' - roughly behind where the Turkish Cyprus Airline offices used to be, more or less in
this area, but I can't remember the directions exactly, I'm afraid
Take all documents, (copies of the other guy's documents will be OK if you are going alone) and either go in, and brave the queues, and 'ticket' system, or grease the palm of the little old lady sat outside, and she will go in and do it for you - I'm not joking, she will just charge you a few Lira.
Change of ownership costs 'in the low hundreds of Lira' - possibly 350 - ish TL.