You've got to be absolutely high if you think Kirei Kotomine beats Kamui. I may have finished Gintama recently, and it has become my favourite show, but Fate is my favourite Franchise of all time. Kirei Kotomine is one of, if not my favourite villain. So, let that rid you of any bias here.
First off, Kirei couldn't even beat True Assassin; one of the weakest Servants in Stay Night. I know he's weaker in Stay Night, and he was in his prime in Zero, but Kirei is not capable of taking on Servants without some sort of advantage (lulling True Assassin into a false sense of security). Don't get me wrong, Kirei is an absolute beast when he was at his full strength in Fate/Zero. His physical feats were not even through reinforcement Magecraft, but through sheer training of his body to perform superhuman acts.
Kamui on the other hand is beyond human. When Yato fight, buildings collapse. Combat between the strongest of the Yato is similar to combat between Servants. The only people we've seen in Gintama that were capable of genuinely beating Kamui are Housen and Utsuro. He's one of the best fighters in the show, and that makes him one of the best fighters in the galaxy. That's when he's in control of himself. When he's hurt or riled up enough, his Yato instincts take over, and he becomes a monster. It took Kagura (his sister), Gintoki (a skilled warrior with a superhumanly indomitable will to continue fighting no matter his injuries - so deadly he was called the White Demon on the battlefield), and Shinpachi (who's above average in skill, but is a normal human) to hold him off. Every punch Kamui makes breaks bones, and shakes organs so much that you might as well be playing basketball with them. He's also got a severe skill advantage, because the fighting skills of the Yato are genetic. Training makes them more skilled, but just simply being alive without doing any training, they're already martial arts masters (the same way babies learn how to walk because it's an evolved trait, Yato are already skilled martial artists because it's an evolved trait).
You know how Kirei is fast enough to keep up with Kiritsugu moving at x4 speed? His little Sister Kagura (who is weaker) is capable of catching bullets with her fingers and teeth. That's already beyond what Kirei is capable of doing. As for strength, Kirei isn't capable of breaking mountains. Kamui is. Kamui is also a more skilled martial artist, combining his inborn fighting skills and a decade worth of fighting on battlefields all over the galaxy searching for stronger opponents to beat.
Now, to give Kirei some wiggle room here, the only advantage Kirei has is that Kamui is not invincible nor invulnerable. Kamui is not bulletproof, and can take damage if hit hard enough. While an average human is not going to be able to hurt him, anyone who is trained and physically strong could punch and hurt him (it probably won't even bother him, but it can hurt him). He's also vulnerable to bullets and blades. Given Kirei could punch so hard that he can crush someone's heart and send them flying 10ft across the room without any Magecraft, he's capable of punching Kamui and actually doing meaningful damage. He won't kill him in 1 punch, or even 10, 20 punches, but he can most certainly kill him if given the chance.
If given the chance... that's the key point.
Imagining a fight between Kirei and Kamui (assuming Kirei is at his best from Fate/Zero and not his weaker Stay Night self), the fight is going to be a lot like Kamui's fight with Sougo in Gintama. Sougo got the crap beat out of him, but he still found these tiny moments in their fight where he actually wounded Kamui several times. That is probably what is going to end up happening. Kirei is going to get the crap beat out of him, but find these little moments to leave some devastating slashes. But if his fight against Sougo, and fights following it say anything, those devastating wounds that Kirei inflicts on Kamui aren't going to make the fight any bit easier. Kamui won't slow down, and won't hit weaker. On the other hand, every hit Kamui lands on Kirei will slow Kirei down. It doesn't matter if Kirei blocks a flurry of punches. Kamui will just hit harder or faster until a punch lands, or Kirei's arms break. Kirei is capable of doing a fist fight half dead in Stay Night, so he can fight on past most wounds, but he won't last. In a 10 round fight, Kamui wins 9/10 times.
Kirei has no amount of Magecraft or martial arts skill to stop someone who is faster, stronger, and more skilled than he is. The longer the fight goes, the more he gets injured, the quicker he dies. The more punches he makes, the more attacks he blocks, the less chance he will have at winning. The only chance he has if he goes all out at the very beginning and somehow makes Kamui underestimate him.
The chances of even that happening are low, because Kamui doesn't go easy on people. He never throws a fight because he likes killing. It's either Kamui is trying, or he's REALLY REALLY trying. Kamui can also sense when his opponent is strong. Given Kirei's calm, cold demeanor, Kamui is going to smell the Executioner off of him.
9/10 times, Kamui is going to have a couple bruises. He'll have some cuts. He may even have 5 blades in his chest. But he'll casually walk it off. Kirei though is going to have half the bones in his body broken, missing limbs, no head, or he could even be ripped in half. He could even very well look like a deflated dummy by the end.
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