BRUSH.
I just happen to own a semi-professional paint sprayer, so it was the way to do the whole barn in about 2 hours - more than half of which was clean up.
I built my hen house later, and did it with a brush. Roller is good, too, depending.
and with the 6x6 sizes you are suggesting, I'd use plywood, not hardie - too many cuts. That's not a bad size to shingle, either. Quick work of it if you have a good hammer. Don't forget your flashing, and don't skimp on the underlay - though you have enough pitch you wont have to use one of the modified bitumen options, which will help w/ cost.
The one caveat with a shingle roof is that, to be effective, the roofing nails have to penetrate the sheathing a good distance - so you will have LOTS of exposed nail points under your roof. Almost 500 nails per square, more if you are in a high wind zone, as I am.
For cost and labor hours, I would not shingle, but I have exactly NO care for appearance, only function. Most aren't so fortunate.