It vaguely sounds like you might be thinking of I, Robot, which was released in 2004, starring Will Smith. Your description is a bit vague, as most of it sounds generic to the "Sci Fi with Robots"; you also report that you remember the protagonist being Caucassian; but imply that you are not 100% sure. As such, here is a trailer, and an analysis of your recollections.
I, Robot (2004)
In 2035, a technophobic cop investigates a crime that may have been perpetrated by a robot, which leads to a larger threat to humanity.
"There was a young man that was in this Science complex. In this science complex, there was a large number of robots that had white smooth plating (their heads and bodies were very well shaped to look like a human)."
Will Smith's character makes several trips to the facility where the robots are actually made, in I, Robot. The robots are designed to look human, and look very much like the example image you posted.
"While in this science complex, there was one robot in particular the man befriended, there was LOTS of hostile robots."
This strongly reminds me of a scene that occurs towards the end of the movie.
The "main character" robot, Sonny, returns to the factory with Will Smith. At this point, the robots have effectively "gone rogue", and all start attacking them. Sonny protects Will, before going off by himself. By this point, Will's character has befriended Sonny, and learnt to trust him.
This would imply that you remember this scene in reverse-order, but you have commented that that may be the case.
"The place he was in was dark, bluish and whitish all in one (to look like a science fiction movie) the drop had a the same sort of colour. The surrounding space was a dull blue grey colour."
The general factory, itself, features a lot of blue and white colours. In fact, a general blueish hue can be found, in other scenes. This sounds pretty generic, however, to the "Sci-Fi with Robots" genre.
"There was a long range camera shot of the man running along a high up path with a rail and a colossal drop on the other side of the rail. .. The friendly robot was a fast runner and the man was trying to get away from it but the robot caugth[sic] up to him, the man said (I think): 'What do you want from me'?
I can not recall any specific scenes exactly as you describe; however, there are several scenes that enforce the physical capabilities of the robots, both in running speed and ability to fall from height. There is one particular scene where a robot falls from a height, and leaves impact scars on the ground, below. What does throw me off is the recollection of the robot chasing the main character; from as best as I can remember, it would be the other way around.
The main robot, Sonny, is accused of murdering his creator and the inventor of said robots; the entire plot revolves around the significant proof, and the fact that robots do not kill humans. There is a scene where Will's character must chase Sonny, who flees during Will's initial encounter.
Naturally, this is again very generic. It is not uncommon for Sci-Fi robots to be shown as having greater speed and endurance than a regular human being.
"I feel like this is when the man was about to fall over a drop and when the robot was trying to either help him up or do something else I can't remember."
This reminds me of a scene where Will's character is drowning, and a robot tries to pull him out of the car.
This might also account for your vague memory of the words "What do you want from me"; Will's character is stuck in a car that has gone into the lake, alongside a child. The robots logic tells it that Will's character has a greater chance of survival, due to being a fully grown adult, and saves Will's character, much to his protests. As a result, the child dies. This serves as the root cause of Will's robo-phobia.
This would mainly correlate to remembering a robot "pulling the main character up".
"When the robot moves, it made this machine like screaming sound. The robot did talk, though I can't remember what it said."
I can not recall any particular sound the robots make, though they did talk. The main robot, Sonny, naturally talks a lot more; but all robots had the ability to talk, and interacted as such.
"I remember there being some sort of blue sheet like wall that looked like a water fall, but made of some material that robots could pass through easily, and the robot used it to get to the man while chasing him (I think the man ws trying to destroy or lose the robot in some way I can't remember)."
This reminds me of one of the final scenes.
Sonny attempts to destroy the main computer responsible for the robots "going rogue", and finds that the main computer is shielded by an energy field. Sonny comments on how his creator made him out of a special alloy, and he never knew why, until he got to the main computer; this special alloy allows him to pass through the force field, in order to inject nanites that would kill the main computer.
I feel I may have missed the mark, as so much seems generic enough to easily apply; and yet, so much seems to bare stark similarity. I may have missed or mis-remembered minor details from I, Robot; but the main aspects are accurate.