Oxford has book, as an intransitive verb, meaning "leave suddenly" or "move quickly". They list it as an American usage, and informal.
Merriam-Webster has book as an intransitive verb, meaning "leave" or "go", especially "to depart quickly". They list it as slang.
FreeDictionary cites American Heritage® Dictionary of the English Language, Fifth Edition to give book as an intransitive verb meaning "to move or travel rapidly". This is the only one that offers any idea of etymology, of those I have found - and it is highly speculative. They suggest it is perhaps derived from boogie.
These definitions, and my experience, suggest that it's very dialect-dependent, though American cultural hegemony has probably made it more recognisable to more people around the world. I've certainly come across pop culture things where characters say "book it" to mean "run away quickly". It is also clearly very much informal. The American Heritage Dictionary version gives the example "we booked along at a nice clip", which suggests to me that it might actually have been more common once upon a time, and didn't have the connotation of leave, just of move. Now, though, the slang usage means it's not likely to be seen as old fashioned, but it is likely to be seen as very informal.