I know the plot of the movie is fiction, but I wondered if that specific part of the story line has a precedent in history. For those who may not know to what I'm referring, TC'S character was a captain in the American wars against the natives. He was then basically rented off to Japan to help them fight the Samurai.
The United States military did in the Revolution. Baron Von Stuben of the Prussian Army wrote the book "Regulations for the Order and Discipline of the Troops of the United States" which detailed marching and other basics of running an army. It was used until the War of 1812.
He also trained the soldiers in all manners of warfare. Bayonet charges, discipline, tactics etc.
He was not the only general that did this. France also sent some generals who were instrumental in raising the US Army, but I'm unfamiliar with them on a level to give a good answer.
Certainly! I know that everyone is answering about Japan but that's my flair area so I'll continue the trend ;).
During the Meiji period Japan hired foreign advisors for everything. Army, Navy, Railways, Lighthouses, Telegraphs and Banks were all built and run by foreign employees who also trained their Japanese successors.
The military was at first modeled on that of the French, but after their defeat in the Franco-Prussian war the Prussian model was adopted. Several Prussian millitary advisors were requested by the Japanes government and Berlin obliged. The most notable of these was Jacob Meckel, who played a defining role in the creation of the Japanese army, writing its doctrine, training manuals and even it's grand strategy. He remains famous among foreign relations experts today for his comment "Korea is a dagger aimed at the heart of Japan" which he made in a letter advocating that Korea be the target of Japanese expansionism (which it later was). He had a stormy relationship with his Japanese counterparts and his rage and petty grudges were notorious. He ended the career of many a Japanese officer by getting in an argument with them and than threatening to resign if he didn't get his away, forcing the government of japan to sideline their own officer in order to keep his (legitimately invaluable) services.
I feel now is a good time to distinguish between "generals" and "millitary advisors." Meckel and his like were millitary advisors, they were not supposed to command troops in war and were merely there to train their local counterparts and give advice. Tom Cruise's character in the Last Samurai is a millitary advisor, and as I recall he finds it highly irregular that he is ordered to command troops in the field (he should anyway, been a while since I've seen that movie). A few times in history, advisors have overstepped their bounds and actively fought for their host country.
One interesting example of this is the character that Tom Cruise's character is based on (well, he was a white person in Japan and that all that matters right Hollywood?) Jules Brunet. Jules Brinet was a millitary advisor for the Tokugawa Shogunate and became rather attached to his hosts. As such, when the shogunate was threatened in the Boshin Wars he took an active role in its defense. Eventually, he would violate orders and lead the last Tokugawa stragglers to Hokkaido and establish the Republic of Ezo there. The Tokugawa loyalists actually went along with Jules Brunets plan for turning Hokkaido Into a French Revolution wonderland with universal sufferage and equal rights. Whether they actually believed in the cause or were currying favor with their powerful French ally is for you to decide. Unfortunately for Brunet, the Meiji government was not goin to allow his experiment to continue in peace. They invaded Hokkaido and crushed the fledgling republic. Brunet and his French colleagues took and active roll in its defense but evacuated to France at last second. One French millitary advisor did stay behind and fight to the end however.
A final interesting case of Millitary Advisors is the Germans in China pre-ww2. Chiang-Kai Sek's nationalists had numerous "elite" divisions which were trained and equipped by German millitary advisors. These advisors exercised tremendous influence over the Chinese army during the early stages of the second Sino-Japanese war. Most notably, their leader, Falkenhausen, convinced chiang Kai sek to fight in Shanghai, the first case of major urban warfare in history. Why were they so influential? Because there so many of them. China earned a reputation as a safe haven for Jews in the German army and other Germans who had beef with the Nazis and large numbers moved to China and became millitary advisors as a way of fleeing Hitler with dignity.
Hiring foreign generals has been done throughout history when local generals aren't up to task for whatever reason. The earliest example I can think of (although I'm sure there are earlier examples) is the hiring of Xanthippus of Carthage by Carthage to defeat the Romans during the First Punic War. Despite his title Xanthippus was a Spartan mercenary. Under his leadership the Carthaginian army was able to defeat the invading Roman army in Africa at the Battle of Tunis in 255BC.
Of course, Sparta almost certainly did not send Xanthippus to Carthage themselves so this was less of a transaction between nations and was simply and example of Carthage hiring a mercenary general.