The ram I shot last year in Wyoming was running. Ordinarily, I try to avoid running shots, but this ram was moving straight at 150 yards and about to gain the cover of an approaching slope. And I was in a solid sitting position, which is how I learned to shoot as a boy. I touched off when the crosshairs were three inches in front of the ram and hit him in the lungs, though about three inches forward of an ideal placement, so maybe a brisket hold would have been better. In any event, it was a one-shot kill. At 150 yards a modern bullet is moving pretty quick, so not much lead is necessary on any animal at that range, with perhaps the exception of a pronghorn moving flat out. Earlier that day I had passed on a shot at a walking ram, but he was at 550 yards with a strong crosswind. Conditions and range often dictate when a running shot can be taken ethically.
Regarding advice to a novice, I grew up shooting a lot of running jackrabbits (and occasionally running coyotes) with a .222. Practice, and experience, is essential. If you have not practiced shooting at running targets, a lot, do not shoot at running rams! They are magnificent creatures and deserve better than a wounding shot.
BTW, my Wyoming ram, at 175 4/8, was recognized by WYOGA as the top ram submitted in 2016. Already this year I have seen pics of three rams taken in the mid-180's. Wyoming is turning out some quality bighorns in 2017.
Good luck in the draws!
HT