Yes that is what it means. A synonym and why they wrote booked is “scheduled”. An actor gets a role and then it must fit in his schedule for filming.

Does booked mean I got a job? - in this context, yes. How is it different from saying, "I got a role, I got a job." - the meaning is the same. The speaker uses the word "booked" because they felt that it was a job they were entitled to. And perhaps to show others how confident they are in their own abilities. It's a kind of 'showing off.' You can book a job like you book a ticket? - yes, sure can: same word - different context - slightly different meaning...