Every model of the solar system seems to follow a specific orientation in which the upper side of the system is Earth's north half and the lower side of the system is Earth's southern half. So is North and South in space, or up and down based on Earth's up and down?
For humans up and down is related to gravity. So you would need to choose a reference first. If you are in outer space and your feet point to the sun and your head to Jupiter, you'd be rightside up (if the sun is the center) and upside down (if Jupiter is the reference).
North and south do not have any physical relationship with up and down. Neither in our planet, nor in space.
In space you have to choose a coordinate system. In those systems you have a reference plane, latitude and longitude. For latitude (+90 to -90) the positives numbers are those that would correspond to the north of the equatorial plane.