Yes, Malta Public Transport operates several bus routes which link the airport to various localities around Malta. The “X” Routes will take you to most localities on the island, with various stops along the way, whilst the Tallinja Direct Routes (TD Routes) will take you directly to your destination.
Further information about these routes is available here: https://www.publictransport.com.mt/en/airport-services
The bus service runs from just outside the Malta International Airport terminal, across the road from the Check-In Hall.
Another bus stop is located on the left-hand side of the terminal next to McDonald’s.
You may also find Malta Public Transport’s official bus schedule here: https://www.publictransport.com.mt/en/timetables