I think if you check out Larnaca airport, you will find that there is a bus/coach service that runs from the airport through to Kyrenia. I'm not sure what the procedure is about passing through the border, but I'm sure your travel agent should be able to advise you about this.
I live in Southern Cyprus, but have never travelled across to the Turkish occupied areas....if I find out any more information, I'll get back to you!
Also, there is no longer a Woolworthsd in Lefkosia ( Nicosia)....this is actually Debenhams now!
EDIT:
Well, it seems you can get a coach/bus to the Ledra Palace, or to the public bus station in Lefkosia ( Nicosia), but from there, I think you will find that once passing through the border, you may heave to get taxis on to Kyrenia.
Sorry, haven't had much time to check this out, so maybe you should contact either Larnaca Airport, or check with your travel agent....they should have some information about transfers, especially as you will be crossing to the Turkish occupied areas. I know that the rules have been relaxed a little, but still not sure what the procedures are! Don't bother going myself as I know it will void my car insurance if I drive over there!!
Why the thumbs down? I'm trying to help!!!!