It's interesting to think about writing about "Rip Van Winkle" from these three critical perspectives. The historical perspective is important, as the story shows the social and political shift that occurred in a short period as the eastern seaboard moved from being a set of British colonies to its own nation. A historical approach would provide the reader with more background on the Revolution and the kind of changes it introduced into American life.
The gender approach would note that the story is told from Rip Van Winkle's male perspective and would question the stereotyping of Dame Van Winkle as a shrew and a nag. What might life look like if you were Dame Van Winkle, depending for financial support on a man who doesn't work much? Is she left to handle everything, from the children to how to make ends meet while her husband loiters around the Inn, talking "about nothing?" Is her anger and "nagging" justified?
A formal approach could look at the language of the text, perhaps with a focus on the imagery (description using the five senses) that Irving uses to create the tone of the story and highlight the contrast between the sleepy time before the war and the bustling energy unleashed in the new Republic. One way to do this would be to focus on the language used to describe the Inn versus the language used to describe the Union Hotel. Unlike historical criticism, formal criticism focused on what is "in" the text.