I've managed to travel from my starting city to a city with a starport - but I despite moving all around the building I can't find a way in - nor are there any NPCs in scanner range (or any the buildings).
Its like "airport" security gone mad.....
You should be able to hit the "Enter" option on your position screen from any point on the facility perimeter and enter it. If it's not coming up, then it's not open to all. I'm not sure if the "NPC transport" on the map is triggered by any starport, or if they have to be NPC owned.
"May the Grace of Ara go with you, and His Vengeance be wrought upon your enemies."