Factual error: When facing the western wall, Poirot orders a man to guard the south gate. The man runs to the left of the frame. The western wall is what remains of the old temple, which lay to the east of the current plaza, so when you're standing looking at the wall you're actually facing east, not west. The man runs the wrong way.
Factual error: Poirot refers to the senior police officer in Jerusalem as the Chief Inspector of Police and implies that he commands the police in Jerusalem. The Palestine Police had no rank of Chief Inspector. The three stars worn by the officer were the insignia of an Assistant Superintendent of Police, the entry-level rank for gazetted officers. The head of the Jerusalem police was a District Superintendent of Police, two ranks higher. As Poirot is famously a stickler for detail, it's certainly not a character mistake.
Factual error: In the breakfast scene involving eggs, Poirot refers correctly in French to one egg "un oeuf" (pronounced "erf"). However, Branagh mispronounces the plural. He said "oeufs" ("erfs"). The correct pronounciation omits the f and the s, correctly sounding like "euhh." Poirot is French-speaking so would know better.
Factual error: The Orient Express had a restaurant coach, four sleeping coaches and two baggage cars, and carried 58 passengers, far larger than the train depicted. There was a conductor always on duty in every sleeping coach and they worked shifts, so more than four would be carried; it's stated that Michel is the only conductor on board.