From Esenboğa to METU
Ankara Esenboğa International Airport is 33 km (21 miles, about 40 minutes) from the city centre. Airport shuttle (named as Havaş) from to the central bus terminal (named as ASTİ) of Ankara are available; Havaş shuttle costs 11 Turkish Liras. Havaş goes to Kızılay, city center and then you have several options. You can use subway(2.75TL), local busses(2.75TL).
From AŞTİ to METU
From AŞTİ, you can use the subway which costs 2.75TL, local busses(2.75TL), or a taxi which rougly costs 30TL.