Generally speaking, you need the article. However, it's hard to say for sure, because it's hard to see the meaning of the somewhat odd phrase 'duty hour'. Who is on duty, and why? Are they just on duty for one hour? Is there something special about this particular hour?
I know a dictionary might not cover all posssible usage of a word but it seems that it isn't a word that can be used in such a phrase, "during duty hour," without having an article before the word "duty. Can you give me a rare situation where it is used as such?