Other mistake: When Ann Darrow is awaiting sacrifice to Kong she is hanging from tight thick ropes around her wrists. However, when Kong shows up he just grabs her and tears her from the plinth. Either the ropes should still be around her arms having been torn from their mounts, or her arms should have been ripped off.
Other mistake: The ladder at the top of the Empire State Building gets loose because a bullet hits one of the screws. It's not very likely that the five other fixtures would snap just because one got shot away.
Other mistake: When Lumpy's head is getting swallowed he lets out a yell which expels his breath. With his head completely covered, including his mouth, he is able to take a full breath and yell again which would be impossible.
Other mistake: In the scene where everyone gets to the island after Ann was taken and the natives hide, Jack goes up stone stairs to where Ann was strung up by the natives before being taken by Kong, Jack says: "She's gone". But when you watch that scene with Spanish subtitles, he says "She's dead" in Spanish.
Answer: I assume you mean, could the building take his weight, not whether an ape would really have the ability to climb a building (if that's what you mean, then it's definitely yes...apes are great climbers). Assuming Kong is proportionally as heavy as normal-sized gorillas, which tend to be in the area of 160kg (~350lbs), then he weighs over 80,000kg (89 tons, give or take). The average human weighs about 62kg, so that's about 1,300 humans, and the capacity of the ESB is over 13,000. So, assuming the building is mostly, or even half, empty while a giant gorilla scales it, the building could handle his weight.
Keep in mind, though, that the weight allowance for the building assumes people on the floors of the building, not climbing on the outside. The outer structure of a building isn't designed for massive creatures climbing on it. While the building as a whole would likely survive, there would be significant damage as Kong would be breaking windows and pulling stone off it as he made his way up.