Continuity mistake: When Ann meets Kong on NYC street, the camera flashes back and forth between them. When it shows Kong, he is surrounded by snow, but when it shows Ann, the street doesn't have so much as a snowflake.
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.
Factual error: Denham is obviously shooting a sound film - he has a sound recordist with him along with the bulky and awkward recording equipment typical for the era, and they discuss the problems of recording dialogue on board. But not once do we see him filming with sound. We see the crew recording dialogue - synchronised sound, recorded on location, which is utterly impossible given the equipment they have and the circumstances under which the fim is being shot. We never see a microphone, a boom pole or a tape recorder. His camera isn't even 'blimped' - soundproofed - and it's handcranked, which makes a racket. They can't be planning on adding the sound later - why have the sound recordist and his bulky and heavy equipment there with them if they are? We see the crew recording dialogue - synchronised sound, recorded on location, which is utterly impossible given the equipment they have and the circumstances under which the fim is being shot. The whole point of post dubbing dialogue is that you don't need a sound recordist in the first place.
Visible crew/equipment: On Skull Island, when Carl says, "No one has lived here for hundreds of years," the distinctive reflection of the reflector screen is visible in the lens of Preston's glasses, as he looks down; there are other moments in the film where this happens with characters' eyeglasses.
Continuity mistake: When Anne comes out of the lake after fighting with the T-Rexes her hair is all wet, but after Kong has defeated the last Rex and puts her on his shoulder to start running, her hair is totally dry. Same happens after she and Jack crash in the river after letting go of the bat. Next shot you see them running towards the temple and her hair is, once again, completely dry.
Continuity mistake: When Bruce Baxter tries to wiggle his way out of having to get closer to the brontos so Carl Denham can get his scene on film ("Shouldn't there be a stand-in for this?"), the gun he holds that can be seen pointing up over his left shoulder keeps appearing and disappearing between shots from the front and back.
Continuity mistake: When Ann is tied up as a sacrifice, waiting for Kong to appear, the position of the skull necklace round her neck differs between the close-ups and wideshot.
Continuity mistake: When Kong holds Ann in his hand, as he sits near the top of the Empire State Building, the way she is positioned in his palm changes repeatedly, depending on the various camera angles.
Revealing mistake: Near the end of the film, when people are running in terror out of the theater and into Times Square, you can see that the people are not leaving any footprints in the snow. The snow must have been digitally added to the scene.
Continuity mistake: In the scene where Ann Darrow is brought to the Venture, there is a wide shot where she is walking and observing the ship. On one of the ramps (the closer one to her), there is a man who is carrying something up the ramp. In the next shot, if you look at the ramp he was on, he has disappeared.
Continuity mistake: When Ann is with Kong on the cliff, her robe and belt repeatedly change position in consecutive shots, before, during and after she dances, somersaults, etc, to amuse Kong.
Revealing mistake: When Ann is juggling for Kong on the cliff edge she starts juggling at one point. If you look closely, you can see that she is simply moving her hands, and the rocks have been added digitally. This is most obvious when she leans right back, and the rocks don't match her hand movements at all.
Continuity mistake: When Carl films in the valley, Bruce has multiple lines of rope round his shoulder which change position in the following consecutive shots.
Continuity mistake: Note the chrome centers on the wire wheels of the taxi that Jack commandeers towards the end of the film. Initially the left rear one is missing, but it magically reappears later on.
Factual error: In the scene where Denham is getting a wide-shot of the valley where they first spot the dinosaurs, Denham's assistant is using a light meter in a brown leather case. This light meter is unmistakably the 'Weston Master Universal model 715/S74'. This model was first produced in 1939. Since the film takes place in 1933-34, this model of light meter would not have existed. (01:21:35)
Continuity mistake: Soon after Kong grabs Ann for the first time and is playing with her the wide shots show him violently shaking her around, enough to snap her neck or at least render her unconscious, but the close-up shots during the shaking show her being much more gently rocked from side to side.
Continuity mistake: As Jack Driscoll flees Kong in a yellow cab through a narrow alley, the view through the (missing) windshield shows a pedestrian in light-color overcoat crossing leisurely at the other end. When the car eventually emerges in the immediate next shot viewed from the sidewalk, that particular pedestrian simply disappears.
Continuity mistake: At the diner, Carl speaks with Ann while she eats, and things on the table change position depending on camera angle.
Continuity mistake: When the crewman receives the message aboard the Venture, he jots it down on a pad of paper. There is nothing under the pad in one shot, but in the previous and following shots there are more papers under the pad.
Continuity mistake: As Ann tries to escape from the T-Rex, at one point her robe acquires a rather large black and muddy stain at her backside on her left, yet in the very next shot that large black stain is gone.
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.