Arrival Airport for Lanzarote is Arrecife.

You can fly to Lanzarote by all low cost Airlines and flights are readily available from all main UK airports.
Flight time is only around 3.5 hours.
Puerto del Carmen is about 7 kilometers from the airport and takes 15 - 20 minutes to reach the apartment.
Best car hire I think is from AutoReisen - their web site is easy to find. Hire can easily be
arranged prior to or on your arrival at the airport.
Full directions will be supplied.
Taxis readily available, cost is approximately 15 euros.