There are no direct flights to Villarrica. To get there, you need to flight first to Santiago and then to Temuco (http://aeropuertoaraucania.cl/en/). We strongly suggest you to leave at least 2 hours between the arrival time of the first flight and the boarding time of the second.
To arrive from the airport to the city, you can take a transfer (a shared vehicle) or a taxi. To go back to the airport there will be a transfer service from the Campus (which should cost about USD 15). You can also call this number +56 9 9362 4194 and ask for a car to pick you up at your hotel or the Campus. (The cost of this service should be about USD 60.)