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.
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.