I'm not sure you are using the term 'statute of limitations' correctly here. I believe you are saying the minor has been charged with possession of alcohol. If that is the case, there is no 'statute of limitations' to apply, it is a done deal. If you are asking how long it stays on their record, then it typically is there for life. You can apply to get it expunged, and typically offenses while a minor of this type are not overly critical in adult life, unless it is the start of a pattern of lawbreaking.
6 mos officers can't hold a misdemeanor charge over a youths head forever.