Yes. The Kentucky Penal Code includes both crimes in the same statute. Depending on what you were doing at the time of the arrest, you could potentially be charged with alcohol intoxication as well as drinking alcohol in a public place.
You may be charged with drinking alcohol in a public place if you consume an alcoholic beverage in a public place, or on any type of public transportation or a public transportation depot, platform, or waiting room. For example, you were hanging out at a public park with some friends, drinking alcohol, and being rowdy. In this situation, you could be charged with both drinking alcohol in a public place as well as alcohol intoxication.