how do you get a gold shinx???

Basically, the "Gold" Shinx is a Shiny Shinx, you'll have to be very lucky to find one, it can take up to 3 or 4 hours or solid battling to find just one of these illusive pokemon. Although, if you have something like an ARDS, you can get a cheat to encounter a shiny Pokemon with 100% chance.

Gold shinx arent gold theyre shiny
The grass will have sparks coming from it, this means that the wild Pokemon will be shiny
This is VERY rare, I've been on the game for at least 81 hours arnd I havent seen a single shiny

It's a shiny Shinx. The odds of getting one are EXTREMELY RARE! Either she stumbled upon it by chance, or someone is a cheater. If your lucky, or have alot of time on your hands, go into the grass where a Shinx is, and run around there for hours eventually, you might run into a gold Shinx. Or just by an Action Replay, then you can get all the Pokemon as shiny, and laugh at your sister, and her stupid Shinx.

The Gold Shinx your sister has is a shiny Shinx. She was very fortunate to run into one of these, because there's a 1/900 chance that the Pokemon you run into will be shiny.

An easy way to run into a shiny Pokemon is to use the Poke Radar, which is obtained after you obtain the National Dex, which is obtained after you've seen all 150 Pokemon in the Sinnoh Dex.

Go into a Grass path, and use the Poke Radar. Go in a patch of grass that was shaking, and battle the pokemon. Once you beat it, some grass patches should shake, again. Go in the patch that's farthest from you, and it should be the same Pokemon you just battled. Keep doing that until about 40 times, and sooner or later, you'll run into a shiny version of the Pokemon you've been running into.

I only have 3 PKMN that are shiney, and I have been playing for 150 hours or so. It is very rare to get a shiney, much less catch it.

The shinx is shiny, to get it is random. It is very rare (about 1 in 8400)