In all the media I've seen the arms basically just stay in place, half bent.

holy shit 6 legged bovine thing

They do the Choo Choo train with their arms.

That paddling thing that puppies do when you hold them above water.

Just windmilling their fists like wrecking balls.

"HMS Centaur, coming through!"

This is the real reason centaurs are almost always drawn with bows and arrows. Not for horse archer vibes, so the artist had something to do with their hands.

But we move our hands for balance as we run, 4 legged animals don't need that. Also how would they time it with the movement of the legs?

Hmm, humans swing their arms while running to counteract angular movement. Presumably, the horse undercarriage wouldn't have this problem, because there's more legs in an offset motion and just some distance between the pairs of legs to hold the body straight. But horses nod their heads during gallop, so maybe the centaurs would do a little chug-chug motion with their arms instead.

I was about to say, "Holding a bow and arrow"

Spank the butt

Immediately what came to mind. It's gotta be this.

Naruto run.

 or 

So you think centaurs are always going to fight someone? That's racist.

If you'd imagine how much something like a centaur would need to eat, they'd probably carry snacks.

One hand scrolls through tiktok and the other holds an iced coffee, just like humans do while driving a car.

 With the whole upper half bobbing up/down like a horse nodding it's head while it runs.

Where'd they get the coconuts?

I think, canonically, they're always going to rape someone.