The Army Captain says that the land he sells is land owned by the US Government, "purchased by President Thomas Jefferson." But Thomas Jefferson purchased the Louisiana Territory, which included land up to Texas and NONE in the Southwest. But this movie takes place in the Southwest (Arizona and/or New Mexico, where the Apache Nation resided), and that land was secured by the results of the Mexican American War (1846-1848) and the Gadsden Purchase (1854), both of which took place long after the death of Jefferson in 1826.
The Army Captain has a closed or hooded holster for his sidearm constructed so the revolver's butt is toward the wearer's back. This would seemingly allow the wearer to easily draw his revolver with his right hand. But cavalry holsters of this era were "reverse draw" holsters, meaning that the butt of the weapon was toward the wearer's front, not his back. This is because the Army considered the revolver to be a secondary weapon, meant to be used by the wearer's left hand, not his right hand, and to be drawn across his body from his "reverse draw" holster. In the 19th Century the Army considered the saber to be a cavalryman's primary weapon, to be used by his right hand, and drawn across his body from his left side (where it was worn). But most people (including, evidently, movie producers) are unaware of this obsolete uniform regulation, and so an incorrect (and non-regulation) holster was inadvertently chosen for this movie.
Geronimo surrendered in 1886 ending the Apache Wars. Therefore, with hostile Apaches, this movie must take place before that date. Yet during the poker game in the saloon the piano player is clearly playing Scott Joplin. In particular, he plays The Entertainer at one point. But The Entertainer was not written until 1902.
In the desert, the rancher has the Cowboy's roan horse and tells Artemis he found it nearby. Yet, in the next scene, the Cowboy is riding his roan horse, far ahead of Artemis.