First of all, the item in the pictures is not a "cartridge", it is a cartridge case. (A cartridge is a complete assembly of bullet, powder, primer, and case.)
We would need dimensions to verify that it is a 44 Henry case, but it certainly could be. The Henry was introduced in 1860, but it might have taken a while to get to Arizona -- you would probably know more about trade on the frontier than I do. The "H" headstamp would indicate that it was manufactured by Henry or Winchester.
The 44 Henry cartridge became technologically obsolete when it was replaced by the .44-40 center-fire cartridge in 1873, but it obviously continued in use long after that.
I do not know how long the rim-fire 44 Henry cartridge was manufactured -- perhaps one of our cartridge collectors might -- but I would guess until sometime in the 1930s, and it was likely possible to find a box or two on a hardware store shelf long after manufacture ceased.
Welcome to the Forum -- lots of good information and friendly people here.
The Old Guy