Pain during the set or does the pain set in sometime after training?
First thing I'd fix is how you get the bar into position. I've always said learning to power clean isn't rocket science, but you're doing some fugly kind of reverse curl and catching the weight way out in front which is sending a ton of stress up through the arm into the shoulder before you even begin pressing.
At the very least get your elbows under the bar, if not in front of them, when you catch the clean. Not this
position, where the bar is well ahead of the elbows when you catch it. And then, when you do stand upright, the bar completely
launches up out of your hand and then you catch it, again, in a bad position. So you're pounding your joints before any pressing even happens.
Second, the bar never actually gets close to your body during the rep. Do you have the mobility to get the bar under your chin? It never comes anywhere close to that and, again, stays way out in front of the body messing up your leverage and creating unnecessary joint stress.
The bar actually looks like it starts out over your toes, which is a terrible position to press from. You can't generate any power and just waste energy to get the bar up in a "straight" line overhead.
Basically, work on totally fixing your starting position. That'll be the first step to solving any other problems.