I guess technically, they mean the same thing. However, the first sentence sort of implies to me that this person works every sunday, while the second sentence in the second sentence this person is just talking about the upcoming sunday. If you were to say, "I work Sundays", this would imply again, every sunday.

Either using "on" or omitting it in this case is all right.

是比较口语化和生活化的用法，不用过于纠结。