Can't watch the video right now because I'm at work. But one neat tip with Rein (not necessarily related to competitive) is;
When swinging at an enemy, aim diagonally at them. In other words, have them on either the left or right of your screen when you swing, not in the middle.
The reason for this is because Rein's melee hitbox (and indeed all melee hitboxes, although Rein's is the biggest and therefore the most noticeable) are cube-shaped. This means you get a bit extra reach at the 'corners', which are roughly at the left and right edges of your screen. This makes it harder for people to simply run away from your hammer.
On the other side of this match-up, if an enemy Rein is chasing and swinging at you, don't backpedal, no matter how tempting it is to back away while taking potshots at him. Instead, turn around and try to run away with your back to him.
The reason for this is that every hero moves 10% slower (ie: 90% normal speed) when moving backwards. Since the enemy Rein is moving forwards in this scenario, he will always catch you up and you will never outrun him if you try to move backwards away from him. (This is also, incidentally, why you can never outrun a Symmetra's beam while backing away from her.)
All of this is assuming there's no Lucio around, of course, as the speed-boost will obviously change this scenario completely.