Bus 191 departs just from the stop in front of the Institute entrance. Only some links go to the airport (2 - 4 per hour), the journey lasts 20 - 23 minutes.
From all links of bus 191, you have to take them for 3 stops to Divoká Šárka (5 - 6 min), and there to change for trolleybus 59 to the airport (the bus departs every 5 - 6 minutes on working days till 6 p.m., and every 7 - 10 minutes in remaining periods, the journey lasts 10 - 12 minutes). The total journey time is up to 25 minutes (plus possible waiting for bus 191 up to 15 minutes)
Looking in the city map, many people choose to walk to Petřiny Metro station, take metro to Nádraží Veleslavín and from there by trolleybus 59. It is a possible way, but not best. Distance to the Metro station entrance is 500 m, what is about 5 min walk, further 2 min to the platform. Metro departs every 4 - 8 min till 6 p.m. (every 10 min in the evening), 1 stop to Veleslavín represents slightly over 1 min, but getting from metro platform to bus 119 represents additional 2 - 3 minutes. When intervals are longer, the trolleybus 59 departures are bound to Metro arrivals from City, not from Petřiny. Trolleybus 59 departs every 5 - 6 minutes on working days till 6 p.m., and every 7 - 10 minutes in remaining periods, the journey lasts 14 - 16 minutes. The total journey lasts at least 25 minutes (plus possible waiting for metro up to 8 min), typically approx. 30 minutes. This way to the airport is slower and includes much more walking than the combination of buses 191 and 59.