Companies that offer the service privately
Transfer from the airport from € 65 / vehicle with driver:
Public transportation from the airport to Benidorm bus station and then hire a taxi to the apartment:
Buses, from € 9.65 / person: https://www.alsa.es/rutas/aeropuerto-alicante-altet-benidorm.htm
With Google, by car from the airport: Route by car
With Google, by public transport: Public transport
Use Google to plan your route from the airport and you can choose the fastest and cheapest.