My guess is that you don’t need to worry.
Mind you: that is a guess. As a practical, matter, you might want to talk to a placement counselor at a school where you wish to attend.
An important issue to consider is that the fact that you have graduated law school may not, in itself, qualify you for the bar exam. You will still need permission from the regulatory authority in the state where you wish to sit for the bar.
Different states have different kinds of regulatory authorities. In Missouri the bar is pretty much self-regulating. In Illinois, if IIRC, one deals with the office of the Supreme Court in order to sit for the exam. I am a licensed attorney in both states.
In 1991 I was friends with a very nice woman who had a very troubled past. Just before she was due to sit for the exam, she received a letter from the Missouri Bar. She was to appear for an interview on the day before the exam began. After this hearing it was to be decided if she could sit for the exam or not. This news came to her just a few weeks in advance, and after she had gone through three years of very strenuous, very challenging, and very expensive education at a university listed in America’s Best Law Schools. Needless to say, she found this kind of disstressing.
She finally found out that she had permission to take the exam (which she passed at her first sitting, and without having taken a review course) on the night before the first day of the exam.
As I recall, there were four reasons given to her for why an interview and a determination were necessary. She had a history of alcohol and illegal drug problems. In addition, about ten years earlier, while she was in high school, she had been charged with assault after attempting to stab a former boyfriend with a knife.
Using that as a standard, I’d say one DUI should not be a bar. It really would be a hell of a world if you could be elected President, but not be allowed into law school.