No, people in Australia do not live "upside down." This is a common misconception stemming from a flat-map perspective.
On a 2D world map (Mercator projection), the Southern Hemisphere appears at the "bottom." Our brains intuitively associate "bottom of page" with "upside down." But the map is a distortion; the territory is a sphere.
Fun fact: The Eötvös effect means you actually weigh slightly less at the equator than at the poles due to centrifugal force, but orientation is unchanged.