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Public bus 200 operates between the airport and Avenida de América bus station in Madrid. It is only 1.50€ and you can use your weekly ticket or 10-trips ticket. Bus 114 does NOT go to the airport terminals and is a 40-minute walk to Terminal 1.
Public night bus N4 goes from Plaza Cibeles to Barajas district, 400m walk from the terminal through a passageway over the highway. Best to get off at the second-last stop, Avenida Central, and then make a right turn immediately after the bus stop. Go straight on, keep going straight crossing the parking lot, and then you make a left to cross another huge parking lot. After that you see the road and the pedestrian highway overpass. Only 1.50 Euros or the 10-trip ticket is valid as well.