There's no train connection with the airport.
Car and bus
Asunción International Airport is situated 9 kilometers north of the city center in the suburb Luque. By rental car you can reach the airport from the city center via the Avenida España and the Autopista al Aeropuerto Silvio Pettirosi, which, as indicated by the name, leads to the airport.
There's also one bus service between the airport and the center of Asunción, city bus 30-A. The ride to the center takes about one hour. Many hotels have shuttle busses running.
Car rental Asunción Airport
You can find a counter of car rental companies in the Arrivals Hall. Five rental companies are present at Asunción International Airport: Avis, Budget, Hertz, National and Thrifty. You can also book a rental car in advance using the services provided by EasyTerra; they compare the rates of several providers in a single quote.
Taxis can be found outside the Arrivals Hall. Make sure the driver turns on the meter. If there's no meter in the taxi, or if the driver refuses to use it, make sure to agree on the price upfront. A taxi ride from the airport to Asunción costs about US$ 16.00 and takes about 20 minutes. (Paraguay's currency is the guaraní, taxi drivers just as easy accept US$).
There's a parking lot outside the terminal. We advise against leaving your rental car at night.