Continuity mistake: Sydney and Nigel are talking with Robert Blaine at the beginning of the episode. When he mentions the Nepalese people there's a close-up on them. One student wearing a white T-shirt is walking up some steps - but in the next shot he is walking down other steps situated earlier in his path. A few seconds later, another flub with an extra; Nigel stops the museum curator just as an extra wearing white clothes enters the frame, but the person is gone in the sudden cut. (00:06:25)
Plot hole: Sydney is able to surmise from the artwork (we could also say from the writing, but her rival is one step ahead of her for 2/3 of the episode and it is established that he does not know the language) the precise location of the koi in Lumbini. The map is 150 years old, but there's no way even with a big stretch of imagination to buy that they both'd be able to pinpoint with such ease and certainty its location in the basement of a random building in the bustling market center of a town, that surely changed plenty during the past century and that does not bear any special landmark.
Factual error: On a lorry for Lumbini, Nigel comments about this being his first day of work. Even assuming that Nigel and Sydney met first thing in the morning and they just took off for a flight that was leaving for India really soon, it is a flight that takes 17-20 hours. Even considering the time difference between "the US" (the actual location is never specified) and Bangalore (closest hub to their Nepali destination), it is really a stretch, since it's impossible that he did not sleep during the flight, and of course it's just hilarious that they'd have a chat about it only after an intercontinental flight.
Continuity mistake: In the prologue, Young Murray finds a man he does not know sitting at a table. He advances towards him in a wide shot of the room, and the man positions his left hand on the corner of the table, but in the following closer angle on him, his hands are close together exactly like in the initial shot by the same angle. It's like the middle one never happened. (00:03:10)
Other mistake: The episode begins with a caption; "Chicago, 1930." In present day, through internet magic (hotel records from early 1900s somehow digitalized and publicly available!) Nigel and Sydney talk about the history (fictional, no actual historical basis) of the Roxbury. Nigel says that the original owner went out of business "after the war", then the company was sold, and the hotel refurbished, despite being, Sydney says, "only 5 years old." Later in the episode, Sydney looks at a plaque from the Chicago Historical Board that dates the elevator as "circa 1901." (00:12:35)
Plot hole: Lori managed to get to the bunker through the elevator because the modern board was still replaced by the antique one. But that implies that she was able to call the elevator as normal. And if people can just call the elevator back but the normal control board has been replaced, then they wouldn't be able to use the elevator as normal, creating quite a problem in a huge hotel. Sydney earlier was not able to switch the board back to the normal one since she went out from a (mysterious?) exit/entrance, so this situation makes no sense.
Continuity mistake: When Sydney Fox retrieves the parchment with the map, she holds it differently (probably turned 180°) between the close-up and the reverse shots. It also should be noted that the world-class archaeologist just casually dumps on the floor like a piece of garbage a ornate XVI century flask. (00:23:50)