Badass Bookworm: Algren is a highly capable fighter, but he's also an excellent tactician (implied to be a major reason behind Bagley recruiting him for the Japanese mission in the first place), an author and a Cunning Linguist.
The Captain: Was a Captain in the 7th Cavalry and is still addressed by his rank.
Manly Tears: Sheds them when presenting Katsumoto's katana to the Emperor.
Mighty Whitey: Averted. Algren is the main character and learns the ways of the samurai, but he's never shown to be any better at being a samurai than the Japanese. The only advantage he has over the Japanese samurai is his knowledge of western war tactics, which comes in handy by the end of the film against the Japanese imperial army.
My Greatest Second Chance: He experiences immense regret and guilt over his past experience with the tribal Native Americans, but the opportunity arises to redeem this regret when faced with a similar situation with the tribal Japanese.
Historical Hero Upgrade: He's an expy of Saigō Takamori, leader of The Satsuma Rebellion, a revolt of ex-samurai against the Meiji government from January 29 to September 24, 1877, 9 years into the Meiji Era. Historically, the reason why the Satsuma Rebellion was so dangerous was because it was an important manufacturing center for cannons. Paintings depicting the revolt show that Takamori's forces had plenty of guns while the well-uniformed Imperial forces mostly only had swords!
Honor Before Reason: He Does Not Like Guns since he considers then dishonorably for one thing. But in real life, most, if not all the samurai ended up using guns later on. He's also clings to the past.
Gentleman and a Scholar: A scholar who makes his living translating the lies of Japanese who never quite tell the full truth.
Quintessential British Gentleman: Though he's more lively than the stereotypical British scholar, a typical trait of whom is generally not a fascination with execution and torture. That said, Algren's vivid description of the process of scalping does seem to scare him straight.
Faux Affably Evil: He comes across as modest and diplomatic when he first appears. Later scenes reveal him to be a Machiavellian tyrant willing to crush anyone or anything who stands in the way of his ambitions.
Jerkass Has a Point: Notwithstanding his greed and hunger for power, he is astute in recognizing Japan's need for modernization and industralization in the context of the expansionism and imperialism of the westerns nations in the Nineteen Century.
No Celebrities Were Harmed: His character seems loosely based on Okubo Toshimichi, a preeminent Japanese statesman responsible for removing the Samuraii's privileged status in Japanese society who (for better or worse) is considered by many to be the founding father of modern Japan.
Pride: He and Bagley are far too confident in their firearms and derisive of the Samurai's fighting abilities - after nearly 300 years of peace - early on in the film. Bagley appears to learn the error of his ways, suggesting sending in skirmishers rather than a main attack force at the start of the final battle with the samurai. Omura has definitely not learned a thing, overriding Bagley and sending his men to the slaughter rather than listening to his paid military adviser.
Smug Snake: Significantly overestimates his own capabilities, and those of his men, and is shown to frequently disregard useful advice, even if it's advice he paid for.
Underestimating Badassery: Is more dismissive of the samurai than any other character in the film. Even when Bagley starts realising what a threat they are, Omura insists on a full attack.
Villainous Breakdown: Omura starts losing his smug cool when the Samurai prove to be more of a threat than he thought, and then he finally snaps entirely in impotent rage when the Emperor disowns him.
Affably Evil: Despite being a slaughterer of women and children (and incredibly racist), Bagley is actually rather polite and makes some effort to be friendly with Algren before he's ultimately (rightfully) rebuffed.
Expy: Reviews have noticed Bagley bears a striking resemblance to Custer himself (who is mentioned in the movie), sporting the same sort of hair-style and facial hair, having served in the the 7th Cavalry, participating in the slaughter of Indians and being incredibly arrogant and racist.
Drill Sergeant Nasty: His method of instilling some sense of order into the new Japanese recruits is bellowing at them in a language they probably don't understand.
Last Stand: Like Algren, likely realises that the first fight against the samurai is going to be a Curb-Stomp Battle, and not in the way they were hoping. Yet when Algren orders him to withdraw with the rest of the noncombatants and advisers, Gant cheerfully responds, "No offence sir, but, shove it up your ass."
The Good King: Katsumoto believes him to be this, until he realizes that the emperor puts his trust in Omura. Truly becomes this after Algren brings him Katsumoto's saber, when he realizes the value of Bushido.