All four alternatives are valid (depending on context), but they all mean different things. Paraphrasing slightly for brevity...
1: If it is possible for him to go, he should go.
Speaker thinks he ought to go (if he can)
2: If it is possible for him to go, he could go.
Speaker is giving a "defining example" usage for possible and/or might (OR same as 4)
3: If it is possible for him to go, he must go.
Speaker thinks he is obliged to go (again, only if he can go)
4: If it is possible for him to go, he might go.
Speaker doesn't know whether he can go or not - but if he can, speaker believes he might actually go. OR speaker is simply giving a "dictionary definition" example usage where the comma can be approximately interpreted as means the same as, or implies the fact that.
Note that although it's somewhat "splitting hairs" to say 2 and 4 are "different", I think at the level of "probable nuance" 2 is more likely to be a "defining example". That's because in BrE at least, could is more associated with capability, whereas might is more associated with likelihood (possible can have both connotations, but in OP's exact context I think capability is a much more natural interpretation).
TL;DR: This is a bit of a silly question, because there is no "right" answer. Worse than that, by setting such a question, the [original] setter is bound to end up misleading students such as OP.
EDIT: The above text answers the original question. It's still a bad test question because we don't know if the second sentence is supposed to be a restatement of the first (i.e. - a "dictionary definition" as referred to in 2, 4 above), or an additional statement (as 1 and 3 above). Also, even if we assume it's a restatement/definition, both 2 and 4 are at least credible.
Doubtless the expected answer is 2 could, because It is possible for him [to do something] more strongly associates with possibility=capability as above.
4 might works better with It is possible [that] he will [do it], where possibility=likelihood.