Update patches for games from the eShop can be installed as long as you have access to the eShop. Your header doesn't matter. It's when you try to go online IN-GAME where private header matters.
That's the great thing about Sky3DS. You can update system firmware and continue on with life and never have to worry.
Here's something else you might like:
You can play OVER-NETWORK games without headers.
Streetpass without headers.
IR transfer without headers.
And your friends can use your Sky3DS as well... so sharing is okay.
When you -DO- put a private header into template, then that game can play online on any 3DS system. Just don't play the Sky3DS and the original game at the same time because they have the same header and Nintendo doesn't like that.
And it doesn't have to be the same game. It can be any game. And you only need one header. Or so I've been told. So pick a game you own, steal the private header, and then use it on all your Sky3DS games.
Good luck! Have fun!