Battles

I would think Jin would be the victor in a match against Mugen. Jin can keep a level head while engaging an opponent and having been formally trained, he knows how to make use of the techniques he was taught by observing the movements and techniques of his opponent. While Mugen's fighting style gives him unpredictability by using random attacks he observed through the various opponents he has encountered, he is a more impulsive fighter as he doesn't think of the best way to engage an opponent and an experienced, trained swordsman would be capable of anticipating his attacks. 
 
A good example of the differences in Mugen and Jin's approaches to fighting would be when they engaged Kariya Kagetoki in individual matches in the final episodes of Samurai Champloo. Mugen's impulsive tactics were easily shrugged off by Kariya who only required a push from the hilt of his sword to send a surprised Mugen into a riverbank. Jin, on the other hand, was able to evade and hold his ground against Kariya, despite his opponent's better skills. Some might argue a better showing of Mugen's battle prowess would be his battle with Shouryu in episode 10. But in that battle, Mugen was barely evading fatal blows from Shuryu's ki-enhanced sword strikes.