El Altet Airport has convenient and well-established transportation links to the city. So the answer to the question "how do I get from the airport to Alicante?" is simple. There are buses and cabs to choose from.
The comfortable blue bus (with free Wi-Fi) of route C-6 runs between El Altet airport and Alicante every twenty minutes from 6:20 to 23:00. The bus fare is 3.85 EUR, payable directly to the driver.
Bus schedules can be found on the website. The bus terminus is located close to the port, near the Melia Hotel. You can also get off at the train station - Oscar Espla stop and at the bus station - Estacion de autobuses stop.
From the airport in Alicante you can also book a cab, easily done directly at the airport. It will cost around 25 EUR during the day and 30 EUR at night.