Since your offense was so recent, it would likely be a problem if you were applying next year. Since you have a little more time, I don't think it will be a dealbreaker, as long as you can establish a record of sobriety and law-abiding behavior. (It may be worth developing a relationship with an NA sponsor or volunteering at a non-profit treatment center or something to shore up your credibility.) Law schools are most concerned about making sure that their students are eligible to enter the bar. I don't know very much about the character & fitness process, but it seems as if people with past substance abuse problems are usually okay as long as there is a good chunk of time between the examination and the substance abuse and they can show that they are no longer involved with drugs, do not face the same temptations, etc.
You also may be able to avoid reporting your plea on some law school applications. Part of the deal with youthful offender adjudication in NY is that it is not considered a criminal conviction. As you approach your application cycle, it is probably worth going over the criminal history questions with a lawyer (your original lawyer may be able to help) so that you know how your plea fits into the questions each school asks. For instance, you may not have to report your plea if the question asks if you have ever been convicted of a crime. However, if the question asks if you have ever been arrested, you probably will have to answer yes and explain the resolution of that arrest -- so you may have to share your status.