The grammatically correct answer is b. “It is important that he come on time” but it is very common (in the US anyway) for people to use “comes” as well.

B - It's important that he COME on time. This is the subjunctive. More information from Wikipedia. [excerpt] The subjunctive in Modern English occurs in a variety of contexts usually relating to desire or conditional actions. Regardless of the subject, the form of the present subjunctive verb that expresses present or past desires (and other uses) in that clauses is the bare form of the infinitive (not preceded by "to"). Hence, the present subjunctive of "to go" is "I go", "you go", "he/she/it go", "we go", "they go". For instance: "It was required that he go to the back of the line" (compared with the past indicative "Everyone knows that he went to the back of the line"); and "It is required that he go to the back of the line" (compared with the present indicative "Everyone knows that he goes to the back of the line"). The English subjunctive also occurs in counterfactual dependent clauses, using a form of the verb that in the indicative would indicate a time of action prior to the one implied by the subjunctive. It is called the past subjunctive when referring counterfactually to the present, and is called the pluperfect subjunctive when referring counterfactually to the past. It occurs in that clauses following the main-clause verb "wish" ("I wish that she were here now"; "I wish that she had been here yesterday") and in if clauses expressing a condition that does not or did not hold ("If she were here right now, ..."; "If she had been here yesterday, ..."). Source: https://en.wikipedia.org/wiki/Subjunctive_mood#English

It depends on the meaning. For the subjunctive, B is correct (though you will hear A.) I'm expressing my opinion of what is important, not talking about his performance. If it is factual that he comes/came on time, A or C would be ok. A. He comes on time. That's an important fact. C. He came on time. That's an important fact. I would be reluctant to use A because it could be confused with the subjunctive. I wouldn't hesitate to use C if that is the meaning that I wanted to convey.

I agree that B is correct and that A is also frequently used. In the US you would hear more A than B.

comes