Douglas Harper's Etymology Dictionary
1940s slang, said to have originated in the U.S. military, perhaps from alleged sexual exploits of Hollywood actor Errol Flynn.
Wiktionary
a. (context slang English) Having achieved a goal or gained access as desired.
Wikipedia
"In like Flynn" is a slang phrase meaning "having quickly or easily achieved a goal or gained access as desired". In addition to its general use, the phrase is sometimes used to describe success in sexual seduction, and its folk etymology often asserts the phrase has sexual origins.
Usage examples of "in like flynn".
He had begun to repeat two remarks that seemed wittier as he got drunker: first, that if you were a connoisseur of Kirschner's bologna and iceberg lettuce you were in like Flynn here, bo, and second, that all assistant professors were like T.
Some of the others shuffled over in striped bathrobes and all of a sudden I was in like Flynn, knowing Slop Chute.