You should find in where you can change your loadout,
So in short, you need to visit the vendor selling stuff, a ship (oyur own or the Apex taxi) or a console, where you have the option to customize your load out. You need to have bought atleast one other space suit, to apply this skin to, as the Flight suit you have by default use the old suit customizations....
So visit a vendor and buy any of the Artemis, Dominator or Maverick suit.
Find the option to customize your loadout, I suggestes to create a new loadout where you equip your new suit,
then selects this loadout and choose an option to customize suit (or similiar)
After some fiddling around you will find yourself where you can customize your suit and apply your bought skin.
I doubt there was ANY qualit control done on this... what seems even worse, is that when I looked at the new Suits skins in the Arx shop, I could not even PREVIEW these skins in the Arx shop, but in the customize loadout you get to see the suits.