How can I beg someone to let me play?

If he won by cheating, I don't think it is honorable for him to win, and he didn't get the opportunity through proper means. If you have any way to prove that he got the chance by cheating, I suggest you reflect this situation to your teacher in charge of this matter and see if you can draw lots again.

But the risk of doing this is that the teacher will not take your point of view and insist on using the result of your lottery, so there will be a gap between you and you should not be given this opportunity.

If not, I think maybe you can invite him to dinner and ask if I can give you this opportunity to show that you really care about this opportunity and want it very much.

But, ha, I think, since he wants to get this opportunity by cheating, then he should not give it to you easily.