However, based on his behavior and the fact that he is the guardian of the Baudelaire orphans, he is likely to be an adult, somewhere in his late 20s to mid-40s.
It's important to note that while he may seem incompetent at times, Mr. Poe is a character who is meant to be frustrating and confusing. He is not necessarily unintelligent or lacking in common sense, but rather burdened by bureaucratic duties and a lack of insight into the dangers the Baudelaires face.