The first time Castle tells Beckett he's going home to deal with Meredith while Beckett goes down to talk to Lanie, Beckett is wearing a brown/gray tweed jacket. In the next scene, in the morgue, Beckett is wearing a bright red jacket with a black lapel. When she gets back to the squad room, Beckett is back to wearing a tweed jacket.
When Esposito tells Castle and Beckett that a witness saw a man duck into a subway station, the camera pans to the station in question and you can see that the sign mentions the B and D line along with the 1 and 9 line. The B & D only connect with the 1 & 9 once and that is on 59th Street and Columbus Circle. Also, the 9 train hasn't been in commission for a few years.
When Beckett and Castle are in the bedroom and she answers a phone call from the precinct, her phone is visible. It is clear and obvious that the iPhone is still locked so Beckett couldn't be talking to Ryan.
At around the 21-minute mark, when Ryan and Esposito walk up to Samuel Schuster, palm trees can be seen the background. There are no palm trees in New York but there are in Los Angeles where the series is filmed.
When Beckett and Castle are in the bedroom and she answers a phone call from the precinct, her phone is visible. It is clear and obvious that the iPhone is still locked so Beckett couldn't be talking to Ryan.
(at around 30 mins) While talking to Noah Kesswood about Michelle, Beckett calls the DEA the Drug Enforcement Agency instead of the Drug Enforcement Administration.
Fingerprints can be changed, or at least altered into unrecognizability. It would be quite a long and painful process, but it's not impossible.