No, the Graduate soundtrack album came first - January 1968. The Mrs. Robinson single and Bookends album came in April. Supposedly Paul Simon repurposed a half-finished song called Mrs. Roosevelt when the movie was due to come out and he hadn’t come up with the material he promised. I don’t know if he knew anything about the Mrs. Robinson character - if not it might explain why the lyrics don’t seem to have much to do with the film other than a general scourging of an unpleasant woman.
Yes, the Beatles song came first, on a 45 where the way John sings it sounds like “coo coo ca-choo”, and I would guess Paul Simon liked the rhythm of it and found a place for it in his hastily scribbled song. (When the Walrus lyrics were printed in the Magical Mystery Tour sleeve, it was rendered as GOO GOO GOO JOOB but who knows if that was Beatle-intended or some transcriber’s fudge.)