There's no train connection with Monterrey Airport.
Car and bus
The airport is situated 15 kilometers northeast of Monterrey's center. The airport is easily accessible from the city center by rental car: You first follow the Avenida Constitución and then the Carretera Miguel Alemán (also: Carretera Federal 54) which leads directly to the airport. The way to the airport is clearly signposted.
From the bus station buses not only leave for Monterrey, but also for Monclava, Saltillo, Nuevo Laredo, Reynosa and Matamoros. There are even international buses, to Laredo and McAllen in the American state of Texas.
There's also a luxury Sky Express Bus between the airport and Monterrey. It leaves from the Valle Oriente bus station on Avenida Lázaro Cardenas. These luxury buses offer free wireless internet on board.
Car rental Monterrey Airport
You can find counters of car rental companies on the ground floor of the Arrivals Hall. Nine rental companies are present at Monterrey Airport: Avis, Advantage, Alamo, Budget, Dollar, Europcar, Hertz, Payless and Thrifty. You can also book a rental car in advance using the service provided by EasyTerra; they compare the rates of several providers in a single quote.
You can book a taxi at a kiosk outside the Arrivals Hall. There are fixed rates for various destinations. A ride to Monterrey's center takes about half an hour.
There are short and long-term parking lots near the terminals. For short-term parking you pay 24 pesos for the first half hour and then 6 pesos per 15 minutes with a maximum of 300 pesos per day. On the long-term parking lot you pay 190 pesos per day.