When you put in a lot of overtime hours at your job, you assume you'll be duly paid. Duly means "properly" or "appropriately."
When something is done duly, it's expected, and it's considered proper or right. If someone gives a great speech, you expect that he'll be duly applauded for it. The word due is at the root of duly, and it comes from the Old French word deu, or "to owe." Something that is duly given can also be said to be owed. Be careful not to confuse duly, which rhymes with the name "Julie," with dully, "without liveliness."