The best price found from United States to Sarajevo is on May 31st, 2017, travelling with Condor. See more options, compare prices and book: search flights.
One hour is generally ample for check-in at Sarajevo's modern but compact international airport, about 12km southwest of Baščaršija.
It has a row of car-hire offices and an exchange booth open 8am to 9.30pm (8pm Sundays). Compare exchange rates with the post-office counter (9am to 5pm Monday to Friday), which also sells BHmobile SIM cards for 5KM. There are several motel-style hotels within 15 minutes' walk and inexpensive Gagana Rooms is just a seven-minute stroll.
On the meter, airport-bound taxis charge around 7KM from Ilidža or 16KM from Baščaršija, plus 2KM per bag extra for luggage in some cabs. However, at the terminal it's not always easy to find a taxi prepared to use the meter. Many ask vastly inflated flat fares. If you start to walk out of the car park, a cab is likely to follow offering a ride to town for €10 all in which is pretty close to the metered fare.
A Centrotrans bus service marked Aerodrom-Baščaršija (5KM, pay on-board) departs from outside the terminal taking around 30 minutes to the Old Town area when traffic is fluid, rather longer at rush hour. However, departures are at most once an hour. Last departures are currently 9.15pm and 10.45pm, though timetables are likely to adapt to changing flight schedules. If heading to the airport, services currently leave Bentbaša at 4.15am, 7.15am and 9.15am with very brief request-stops at Vijećnica and Baščaršija and Katedral bus/tram stops a couple of minutes later and potentially at all bus stops along the tram route as far as Otoka. You'll need to watch carefully as the bus barely slows down and only stops when specifically signalled.
A more frequent and cheaper alternative from the airport area is to take bus 31E or trolleybus 103 (1.80KM), which run every few minutes till around 11.30pm. The nearest stop to the airport is Konzum Dobrijna, around a 700m walk from the terminal. To find it, turn right as you leave the airport car park, then left at the large 'Hotel' sign. Shimmy right-left-right past Hotel Octagon, then turn third right at the Panda Car Wash (Brače Mulića 17). Before the Konzum Hypermarket (Mimar Sinana 1) cross the road and take trolleybus 103 heading back in the direction you've just come.