The sentence is a bit ambiguous, but it seems to suggest that the person is typically very punctual and arrives right on time, but perhaps on this occasion, they arrived so exactly on time that they were almost late. In other words, the person is usually very reliable when it comes to being on time, but in this instance, their arrival was so precisely timed that they may have cut it very close to being late. So to answer your question, the person being referred to is someone who is generally punctual, but perhaps not in this specific situation. The sentence does make sense, though it may require a bit of interpretation to fully understand the intended meaning.

It's just extremely informal speech. The idea is that everybody else arrives at least a little bit early, so when he walks in at exactly the appointed time he's the last one to get there.

It's a joke, for example, if he has to be somewhere at 8:00, he's so 'on time' that he will arrive exactly at 8:00 such that if he arrived any closer to the exact time/later, he'd be late. Usually a more responsible person would arrive several minutes earlier. You can picture this person coming through the door for a meeting or class at the exact time it is scheduled to start, rather than arriving several minutes beforehand. The phrase 'I'm that guy' is just used to casually express something about yourself. Or if for female, you could say 'I'm that girl/woman who... has never missed a plane, because I always arrive at the airport 3 hours early,' for example