When Castle is getting ready for his date with Amanda Livingston, he ties his tie twice while talking to Alexis.
(at around 55 mins) When Beckett points at the spider on Castle's shoulder, Castle looks down at it, and in the next cut he looks at it again.
As a seasoned detective, Beckett would not have revealed details about an ongoing investigation to her date, a total stranger.
Castle says, "They're gonna forget all about you and I." As a professional writer, he should know that that should be phrased, "They're gonna forget all about you and me."
(at around 23 mins) When Alexis is complaining about Castle's presence on the Most Eligible Bachelors list, she mentions a friend who is "... seventeen, but she'll be legal in three months."
The age of consent in New York is seventeen, not eighteen.