Upvote:1
City buses 36, 37, 45 and 47 stop at the airport. Information on their routings is really poor though, but you need to travel in the direction of the left (from your POV when exiting the airport), and ask for the closest tram stop ("tramvajski stajalishte").
From the closest tram stop, catch tram 3, 4 or 6 to wherever you need to be.
Alternatively, a simpler option is catching a taxi (Crveni taxi is recommended - NEVER take Sarajevo taxi, which, although the best option in the city in general, is legit mafia at the airport) to the Stup tram stop, from where trams 3, 4 and 6 run to the centre. Make sure the driver uses the meter - if not, insist on it.