Maybe someone has an answer to this question... there were cap boxes and cartridge boxes. The infantry cartridge box was very large for pre-rolled musket cartridges. For a cavalry man, the cap boxes were rather small, about large enough to hold 2 packets of combustible ammunition but whether the ammunition was carried in a "cap" box or whether there was a small, special cartridge box just for a cavalry man- I'm not certain. I've surfed the net but there is all sorts of images that could simply be wrong.
To the best of my knowledge, for a cowboy, you probably want to consider a slim Jim type holster worn cross draw. The cross draw was easier to pull out the revolver when on a horse. On a bag/flask. etc- never say never but I can't recall seeing many photos where such was worn. Where would a cowboy carry combustible ammunition? Good question. We normally think that pockets are post 1840 but that's not entirely true, there are Capots with pockets, some drop front pantaloons, etc. Maybe a vest had a pocket for combustible ammunition.
And....since this is all about trying to replicate what was, might as well figure out the pc way things were done. That's half the fun.
For example, I made some replica Colt and replica Sage wood boxes complete with gummed labels and pull string/wires just to experience how fast a person could rip open the label, stuff the rounds (clear/unfired revolver in this case) into the cylinder, cap and fire. I compared that to withdrawing fired cases from a 1873 Peacemaker and reloading. The old cap and ball was about equal in time.
I think it was one of the Wilson pictorial books- "gunfighters of the Old West"?- that mentioned the town fathers in Abilene agreed to reimbuse Hickok for ammunition for target practice. I think he was still using the foil type cartridges and shot up about 6,000 rounds. In any event, they were very common.