I'm not sure I understand. Usually one simple DUI conviction does not lead to a denial of naturalization, so I wonder what else is going on. I would need to see the denial notice and your N-336 appeal form. You should at least consult with an experienced and competent immigration attorney if not hire one to represent you. If the appeal is denied, you will need to assess whether and when to reapply.
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Not only is the above correct, but wherever any arrest history is involved, it is imperative to obtain and provide the original Certificate of Disposition - you should certainly get this before the date of your N-336 hearing, and it would be better to obtain it before you meet with an attorney.
While a single offense of simple DUI is generally not considered a Crime Involving Moral Turpitude( or"CIMT", which would be grounds for denying an application for citizenship), failure to disclose and document any arrest history as a part of the N400 application would be deemed fraud, which is definitely grounds for denial. If there was more than one instance of DUI/DWI, or if there was a DUI charge with any complicating factors such as driving with a suspended license, or with a child in the car, then it could be a CIMT. You will need the original records pertaining to the arrest and disposition of any charges, and you should discuss the particulars of your situation thoroughly with an attorney before your N-336 hearing.
This is general information only; it is not intended as a substitute for legal advice, and does not create an attorney-client relationship.