Yay Alexis! I'm pretty happy with Alexis. The rest of the episode? Well... let's go over what happened.
Castle and Alexis are on their way to London for Castle to attend a writing conference. Alexis announces that she has plans of her own when they get there, and Castle is upset that Alexis doesn't want to spend the whole trip together. After some turbulence, which makes Castle nervous, he notices that the flight attendants look nervous. He goes to ask them what's wrong. The pilot tells him that the air marshal aboard the plane seems to be missing. Since Castle works with the NYPD, he's the closest thing they have to law enforcement. Castle, along with the flight attendants and Alexis, sets out to solve the mystery.
First of all, they find the air marshal, dead, down in the cargo hold. Their first suspect is a man who has links to terrorism. However, this proves to be a false lead, as he only sneaked his way onto the plane so that he could get home to his ailing mother. He's on the no-fly list thanks to his cousin, who was involved in terrorist activity. Next, they track down the air marshal's phone and discover texts that lead them to another passenger. This woman seems to be the logical suspect- and sure enough, they find the air marshal's gun in her bag! Her story is that she was the air marshal's girlfriend, and she was convinced he was cheating on her. She disguised herself and got onto the plane to see what he was up to. However, when they tell her that the air marshal is dead, she seems completely shocked. Castle finds himself believing her.