In the beginning of the movie the Last Samurai Tom Cruise uses a revolver as his primary weapon instead of a rifle? Would this be common in the 19th century in American history?
In the 19th century and earlier, officers didn't carry rifles or muskets for two reasons.
One, particularly prevalent in European armies, was that an officer was also a gentleman - an idea reinforced by the fact that officers were not technically enlisted in the army but granted a commission by the government to command troops - and thus should not carry a long-arm like a common soldier. Officers wore swords, as both a symbol of their rank and for personal defence. Many officers augmented this with a pistol, feeling that having some form of firearm was a good idea.
As revolvers became reliable to the point where an officer could use it instead of, rather than as a supplement to, his sword, officers of many armies ditched the sword for active service but kept the revolver. This was aided by the fact that officer corps were, by and large, in a process of professionalisation, and the social rank of an officer was less important, which meant that the sword as a symbol of rank could be reserved for parade occasions.
The second reason was that an officer's primary weapon was his men. In most situations in 19th-century and earlier warfare, an officer wasn't supposed to get involved in the melee or the firing-line, but hang back and direct his soldiers, assessing the entire situation, not getting sucked in to fighting. If he needed to defend himself, then a sword or pistol would be adequate for this until his men could move to support him.