Store eggs upside down or not?
By "it stays longer" do you mean that the egg stays good longer or the membrane/air pocket stays around longer?

I don't know why you would care about the air pocket so I'll go with the other.

Orientation is not going to noticeably affect egg quality. I don't use egg cartons. I store my eggs in plastic tubs (carefully). The eggs end up in a random orientation and there is no noticeable difference between any two eggs.

I have heard to people storing the eggs differently to center the yolks in preparation for boiling. This is purely a cosmetic thing.