The story is interesting, but it's obvious it's fake. For starters, you wouldn't just happen to find yourself sitting next to Einstein in 1955, and it only gets more improbable after that.
This story was originally published in 1955, the night he met Einstein didn't happen then. I didn't see an exact date but he did say "When I was a young man" so I assume it would have been quite a bit before 1955.
Presumably this happened sometime between the late 1930s–early 40s, when Weidman (b. 1913) was a “very young man, just beginning to make my way”. Per Wikipedia, Weidman’s first novel was published in 1937.
Had to be after 1931, when the Crosby song (the title of which he gets wrong) was recorded.