Do I need to repeat a verb in a second clause?

Either sentence seems to convey the meaning in a doubtful way and for that reason I'd use the following construction.

• Not only will he pass the exam but he will also excel in sports.

They are both grammatical. Which to use is a matter of style.

When you use ellipsis, the thing that comes before both parts does not need to be included in either part.

However, what should come before is also up to interpretation—as is if it should be fully or partially used:

• He will ^{not only pass the exam} but ^{[also] excel in sports.}
• He not only will ^{pass the exam} but ^{[also] excel in sport.}
• He not only ^{will pass the exam} but ^{will [also] excel in sports.}

Note that I have put also in brackets because it could be considered redundant. That's not necessarily bad, but it, too, is an optional part of the sentence.

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However, ellipsis does not need to be used in any way at all.

In fact a version of the sentence that uses no ellipsis at all would not drop the repetition of the pronoun either:

• He will not only pass the exam, but he will also excel in sports.

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All of those versions are possible.

So, no, you don't need to repeat the verb (or the pronoun). It's a matter of personal choice, and whatever style guide that might be followed, if you do or not.