The minibuses for Bilkent University leave Sıhhiye every half an hour and they have a stop at AŞTİ (Ankara Bus Station). You can go to AŞTİ (Ankara Bus Station) by Ankaray (Ankara Metro) from the city center, then from AŞTİ you can get to Bilkent University by the same minibus departing from Sıhhiye.
The cost is 100-120 TL (30-40 USD) to get to UNAM by taxi from the airport.
For 10 TL (3 USD), the Havaş Shuttle Bus Service will take you to AŞTİ (Ankara Bus Station) where you can take a taxi or minibus (see above) to reach Bilkent University.