Palal is right, you won't pass by the airport. The airport is located on the Bucharest to Sinaia/Brasov road on the northern side of the city. The motorway linking Sibiu to Bucharest (or to say it better: the motorway from Pitesti to Bucharest as the stretch from Sibiu to Pitesti is a twisty provincial road) will enter Bucharest from the west. So definitely drop it off in the city if you are not heading straight to the airport!
If possible, try to avoid driving into Bucharest during late afternoon/evening rush hour (5-7pm on weekdays or Sunday) as it will be crowded (it's one of the worst cities for traffic jams in Europe, and driving style of people can be rather erratic just like in Italy so drive defensively if you are not accustomed). The main roads (Bucharest-Sinaia-Brasov-Sibiu-Bucharest) are all fine although the amount of double carriageways are limited so always add an half hour to an hour to Google Maps driving time just in case if you need to be somewhere on time. On minor roads always be vigilant for potholes. Apart from that, it's not as bad
