is it possible to buy a cheap mobile phone sim card here ideally with a bit of free credit on it for use during my stay?

Yes Robonz. The Vodafone shop can sort that out for you... as can the Infomatica shop. Both are almost opposite the Post Office in Playa Blanca. You may need to already have your phone unlocked though.
This map uses an envelope to show you where the Post Office is (signposted as Correos).
http://www.lanzarotelates.com/pb_map.php

http://www.gandy-draper.com