There used to be an artesian well in Rose City. It was at the general store, which I think is closed now, but it was just out in front of it. Its been awhile since I have been up there, but I think the well might still be running. If it is running, probably no way to hook up a hose, so you would need jerry cans or something to add it. Or maybe just hold the hose over the pipe?
That said, you would probably get off at 202, and then just straight down m-33, and if it was there, it should be just past the shell on your left. I think there is a small bar right before what used to be the general store.
The thing is, thats a pretty hefty jog out of your way. I suppose you could go up to mio and cut back across 72 into grayling.
I would probably at least get off at 202 and ask in the gas station right there. If they can't suggest anything you could always try again in west branch.