From Alicante airport you can take a taxi directly to the flat: It takes about 25 minutes and costs around 35 euro.
Alternatively, take the airport bus to Alicante (costs about 5 euro), get off at Luceros or Mercado and then go underground to take tram Line 3 (yellow) to Muchavista (cost 1.5 euro, machines inside the tram). That takes about an hour in total.
A third option is to hire a car. It's pretty cheap to do so (Gold Car is my favourite). Parking outside the complex is pretty simple.
The flat is not far from Alicante, El Campello, or even Benidorm. And excellent transport links are in place. You don't need a car if you really want to get away from it all.