How is the universe bordered? This answer says that some models describes the universe as finite. How do those models describe the universe's border?

Does the border (theoretically) exist?
Is it a solid border? Do they predict what happens if matter crosses this border (maybe it simply can't)?
Your intuition is correct.  It does not really make any sense for the universe to have a border.  Would it be an unbreakable wall? What would be on the other side?  Models with a finite universe would not have a border.  The universe would be curved into a closed loop like the surface of a sphere.  In other words, if you go far enough in any one direction you come back to the starting point.