Hi Jolene,
I have used this type of auger somewhat extensively (worn out at least one in caliche soil in AZ), and that particular one is a 3/8in shank. I suspect it would snap after the first rock or two. Also, these depend on your drill, so the best augur on a wimpy driver will disappoint. Then, there is "rocky" and there is "rock". My property has rock, and I use a rotohammer if I have to put electric fence posts into it. Some rocks in some (relatively loose) soil will respond much better to a big auger like that.
You might do a sort of test on the soil, with a trowel augered down vertically to the depth of the desired hole - if you can make a trowel do it with some patience and effort, an auger like this will do it with less effort and faster. If you think the soil would still be good to drill, I'd first look for a 1/2in shank on the auger, as long as you have a 1/2in-chuck drill. 4in diameter is big. If you want a big hole, perhaps drill a 2in diameter hole first, then a 4in to break in the sides, but that depends on your posts and the needed hole size. Remember that your hands have to react the forces, and a 4in auger will kick hard on a rock. It's like a gear ratio. I've tweaked my wrists doing this, and I am pretty careful with tools generally. Brace the drill with your leg.
As far as a suggestion,
this one is smaller diameter and looks substantially easier to use while standing, with a 1/2in shank. That's not an endorsement, but it ticks the boxes I suggest above.
A
digging bar is another option, will not flinch at rocks, but it's heavier. If you have physical limitations, it might be more difficult to use than a drill (though the drill will hit you hard when the auger hits a rock!). It has the advantage of not needing batteries, and probably lasting for decades. You might find one at your local hardware, too. You can get pointed tips that would be better for just the fenceposts, or flat tips for digging.
Best of luck!
Mark