Why did Robert Fischer take the plane with Cobb's team in Inception (2010), and how did he get sedated on the plane?
This is actually explained in the film. First they find out that he routinely takes the flight Sidney - Los Angeles. Then they cause trouble with his private jet so he has to fly with public transport. And Saito just buys the whole airline for them to have all the freedom to organize being his co-passengers and have flight attendants they can trust:
Saito: Sidney to Los Angeles, one of the longest flights in the world. He makes it every two weeks.
Cobb: He must be flying privately.
Saito: Not if there were unexpected maintenance with his plane.
Arthur: Would have to be a 747...In a 747 the pilot's up top and the first class cabin's in the nose so noone would walk through. But you'd have to buy out the entire cabin and the first class flight attendant.
Saito: I bought the airline...It seemed neater.
Cobb: Looks like we have our 10 hours.
And how Fischer got sedated is explained, too, though a bit more visually. First Eames steals his passport in order for Cobb to get him into a conversation later. Then Cobb hands him his drink and secretly slips some drug into it. Now if this is the final sedation or just something to knock him out until they get him to the dream machine (that comes with its own sedative) is not clear, but it certainly was to get him sleep away for the next steps of the heist.