Continuity mistake: Carl is talking to Ann in a restaurant. He asks her if she's a size 4 and she moves to stand up. In the background is a waitress walking toward them. As she stands, the camera changes and the waitress has disappeared. (00:14:50)
Continuity mistake: Just before the wall is spotted, Jimmy is in the open crow's nest reading "Heart of Darkness". He puts the book down and stands up. In the next few shots, the book has completely disappeared. Then the ship takes a pounding for 3 or 4 minutes. But a few minutes later there's Jimmy, once again reading the same book. (00:46:05 - 00:54:15)
Continuity mistake: When the "headsman" native is shot by Captain Englehorn, he throws his arms in the air. He repeats this gesture with the second shot as well. (01:01:20)
Continuity mistake: After being offered to Kong by the natives, it appears that Ann is wearing pantyhose for a brief moment. In all scenes on the island before and after this, she is barefoot. (01:15:00)
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: After the running with the dinosaurs scene. Driscoll and Baxter are talking near a swamp. In the first shots of Baxter, he has a coil of rope over his shoulder. We see a shot of Driscoll and when it goes back to a shot of Baxter, the rope is gone. We never see Baxter remove it or see the rope laid down as he walks away. (01:28:50 - 01:29:20)
Revealing mistake: When Carl is standing next to Kong's unconscious body saying, "His name will be in lights on Broadway," look at the Venture in the background. Even though it's not stuck on anything, and despite the choppy waves, it doesn't move ANY. Even if anchored, it would still bob gently in the waves; instead it's frozen solid. Obviously green-screened, or possibly a matte painting. (02:17:40)
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.
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.
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.
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.