They are both right but the first is more natural and commonly used.

Both of these sentences make sense. 'Except' is perhaps not the best word to use here, as it suggests some kind of exclusion. It's not incorrect, though. 'Besides' suggests 'in addition to'. However, to me the most natural way to say this would be: Nobody knows apart from him.

It's more natural to say "Nobody knows but him." or "Only he knows." In your sentences, though, both are correct. Just be careful, because the meanins CAN be different. These days, 'besides' is mostly used to begin a sentence that gives additional, but unrelated, information about a previous subject. For example: A: Do you want to go a baseball game this weekend? B: I can't make it, I have to study for a test. Besides, I don't like baseball. In that case, it has a meaning like 'also'. 'Except' cannot be used in that situation. Similarly, you could (rudely) answer by saying: B: I would, except that I don't like baseball. Here, 'except' has a meaning similar to 'but'. You cannot use 'besides' in this situation. You are right to be confused about these two words, but in your example, the meaning is the same.