book on

book (one) on (something)

To reserve a spot for oneself or another on a mode of transportation, such as a train or plane. My flight was canceled, but someone at the airline was able to book me on another one that leaves today, thank goodness.
See also: book, on

book someone on something

to reserve a place for someone on some travel conveyance. They booked us on a direct flight to San Juan. He booked himself on a flight to Manaus.
See also: book, on