I like playing guitar outdoors and I'd like to go out to some spots other than my front porch and need to use my four wheeler to get there. What would be the best way of transporting an acoustic guitar hard case on an ATV?
Very carefully, I think...

That said, without knowing the make, or tentative luggage attachment points, on your specific 4 wheeler, I can only suggest that you get some thick foam sheet, (4+ inches), cover it with a waterproof material, and sandwich it in between the luggage rack, (?), and the case, with the usual bungee cords as attachments. Perhaps a home bed pillow, given a similar cover, could also work. If you're going to try something like that, please insert a locking cable direct from the case handle, to the frame, in case the load shifts.

Plan "B", leave the case home altogether. Many soft shell "gig bags" have straps, which allow them to be used as a backpack.

Above all drive carefully!
Gigbag with backpack straps, and you'll be able to put it on your back with no worries. Get those that are semi hardcase if you're worried. A pure soft padded case might not protect.

Or, try the Gator G sling. Might work too.