Car hire in Lanzarote is relatively inexpensive and if rented from airport to airport the saving on transfer costs to Playa Blanca can, dependant on car type and season, provide a significant amount of the car-hire cost. While car hire may not be absolutely necessary for Playa Blanca, dependent upon where you stay, it would also give you the opportunity to explore the rest of the island. If you do hire a car use one of the good local companies such as Cabrera Medina – they do list a few automatic cars if this is an issue.
Some to avoid too like Gold Car and Orlando– see this thread best-car-hire-company-from-the-airport-t7881.html
If the map shown on the VRBO web site for the villa you mention (?? Casa Lucia) is correct on Calle Inglaterra then it appears from Google maps to be some way back from the central portion of the resort and probably a bit more, but not impossible, than I would consider walking to get to the town restaurants etc.
Parking is no real problem in Lanzarote, except perhaps in Arrecife, and the island is well signposted.
Playa Blanca is a long resort spread from the Lighthouse in the west to the Papagayo beaches at the east. The centre portion around the old town and ferry port area has plenty of restaurants, bars supermarkets etc along with “tourist” type shops and with small beaches in the town plus Flamingo and Dorado beaches at each end on the outskirts. There is a smart new (about 10 years old) marina (Rubicon) at the eastern end with shops and restaurants and beyond that the magnificent, but fairly isolated, Papagayo beaches. The western end beyond Flamingo beach has a lot of villa developments back from the front and mainly Hotels along the sea-front. The are some small areas with a few restaurants but no “centre” like the town or the Marina.
I believe there are buses that run around the town together with bus routes to other towns. There are also taxis fairly readily available.
Enjoy your trip!