Or rather Alghero-Riviera del Corallo, as it says at the terminal. Whatever name you prefer, it has everything an international airport needs: a bar/pizzeria, shops (including a pharmacy), car hire, a bureau de change, a cashpoint, a children’s play area, free Wi-Fi and a tourist information point.
Connections to Porto Torres are courtesy of the regional bus company ARST, with two trips a day; you can buy the €3 ticket at the automatic machine inside the airport terminal or, for an extra €1, on board the bus. The trip lasts about 40mins and the stop is outside the airport. If you want to go to Sassari, the ARST bus stops inside the airport area. There are nine 30mins trips a day and the ticket costs €4 (€5 if you buy it on board). From Sassari you can get to Porto Torres in 30mins for €2 (€3 if you buy your ticket on board).
A taxi ride between Porto Torres and the airport will set you back about €50. The trip takes about 25-30mins.