Mugen chases after Yatsuha with his arms spread and makes the sound of a plane engine but he shouldn't know what a plane is.
No matter what Champloo dating system you like, there's no way he could have this gun. Though the first large war guns with a revolving cylinder (needing to be turned manually) had been invented in 1718, the first patent for a revolver handgun was awarded to Colonel Samuel Colt in 1836. Before the famous Colt revolver only one and two-barrel flintlock pistols had been invented for hand held use.
Just as Mugen bears a sword and his rival Mukuro bears a gun, in Cowboy Bebop (Watanabe's previous work), Spike bears a gun while his rival Vicious bears a sword.
Fuu's diary note
Elise suggested that the little "x" shaped character drawn next to "baka!" on this page is meant to resemble the veins standing out on anime characters' faces in anger, which would be Fuu making an anime reference many, many years before the existence of anime.
This would-be hotshot swordsman claims to be so fast that he's known as "human electricity" (rendered by his VA in Japlish exactly that way, but in some translations given, less anachronistically, as "human lightning"). Not only is Seishirou not nearly fast enough to be called anything of the sort (in fear of Mugen he does admit he made it up himself), but, though it's not impossible, it's not too likely that he'd use this terminology. The word "electricity" was coined in England in 1600, but there was no electrical anything in Japan until long after that. (Electricity was first used in Japan on March 25, 1878 at the Institute of Technology in Toranomon, Tokyo when an arc lamp was switched on in commemoration of the opening of the Central Telegraph Office).
Mugen reveals his tanto (short sword) for the first time. It was concealed in opposite end of his scabbard.
The jogger who passes Mugen
Not only would this cheerful fella almost certainly not have been running just to get exercise, a concept that didn't exist yet, but he would definitely not refer to it as "jogging", a term first applied to athletic running in the late 1960's & 1970's. Though running as a competitive sport goes back as far as the concept of sport itself, it wasn't introduced in Japan until after the Meiji Restoration (1868) along with other Western sports such as baseball. (Up until then, Japanese competitive sports were all derived from warrior skills: archery, kendo, jujitsu and the like).
The revolutionary rhetoric and the importance of marijuana to the plot are certainly intended to make one think of the general climate of America's 1960s, though there are no direct points of comparison I can see. However, some have pointed out that the souhei master's inflammatory speech is reminiscent of crowd addresses by the likes of Malcolm X and Che Guevara.
Marijuana has been used as an agent for achieving euphoria since ancient times; it was described in a Chinese medical compendium traditionally considered to date from 2737 BC. Its use spread from China to India and then to N Africa and reached Europe at least as early as AD 500. A major crop in colonial North America, marijuana (hemp) was grown as a source of fiber. It was extensively cultivated during World War II, when Asian sources of hemp were cut off.
Nagamitsu and his Kago
He's carried everywhere in a palanquin, or Kago. No way would he have been allowed to do this: those who may ride in palanquins are all persons of distinction who are connections of the Tokugawa clan; lords of domains (Kuni) and lords of castles having 10,000 koku (bushels) and upwards; the sons of provincial Daimyo (beneficiaries), lords of castles; chamberlains and higher functionaries, and the legitimate sons of such (i. e. sons by their wives; but not sons by their concubines); persons (of any rank) above fifty years of age; of the two professions of doctors of medicine and soothsayers (astrologers, onyoshi) and invalids and sick persons. Apart from the above named, irregularities must be prohibited; but those who have applied for and received official permission to ride are not included in the prohibition."
Nagamitsy wears purple (the Imperial color, forbidden to all but royalty) and detailed with a flame-colored pattern that makes him look as much like a custom hot-rod as his palanquin.
Tunde Olusanya thinks this: "The Gangster, who originally had the opium (the bad guy for the day) was to me an unmistakable reference to a satire of Bugsy Malone and the gangsters of his time, an example of this satire is present in the old Bugs Bunny cartoons where he typically went by either Mugsy or Rocky, always followed by a bulky but quite stupid goon. This character Mugsy is a classic mobster from the early gangster years of Chicago. Thick black eyebrows, dark skin balding with the remaining hair thick and black, always chewing on a lit cuban even when talking...The badguy in this Champloo episode had all these characteristics: very short, thick black eyebrows along with the trademark furious face, short-tempered, always chewing the large cuban looking cigar (come to think of it did they smoke those kind of cigars in the 1600's?) [Tunde is seeing things here: Kogoro smokes a pipe, not a cigar], fully equipped with two goons, even had the classical shot of him yelling at his goons on two instances."
This is the first episode to call attention to Fuu's sidearm, so it's a good chance to discuss it in detail. This cute pink tanto with its dangle of little ornaments is both Edo Period and 21st century; it's as decorated as any kid's backpack or cellphone on the DC Metro. Not only is the convention of "pink for girls" completely modern, but the decorating of one's portables with cute little dangles has only been a major craze in the past four years or so. [Peter Payne from Japan, however, points out an important historical precedent: the Japanese use of carved bone or wooden figures and charms called netsuke, which were traditionally used to secure the drawstring of a pouch holding money or valuables, dated to the 17th century.
In this episode we find out that Fuu is fifteen years old.
The overhead lamps in the gambling den
These do not flicker and do not cast the same sort of light as the candle sconces which are also nearby. If they're oil lamps (which they seem to be in the one shot from underneath them we get in episode 4) I can't figure the closed shades; but it's really not likely that they're electric. Overhead chandeliers came into vogue in Japan in the Meiji period, but not electric ones... See anachronism notes on Episodes 15 and 19 for the idea that gaslight may have come into existence earlier in the Champloo universe than in our own Japan.
Yakuza boss Rikiei's shades
(seen also in Ep. 3). Might just as well be Oakleys. Neko-san points out that Rikiei's glasses are not only stylistically wrong for his era, but technologically unlikely as well: "Irregularly-shaped lenses (Jin and Reikiei both have them) are also a bit too modern, because in those days lens-grinding was pretty difficult and time-consuming. Irregular lenses are not easy to center properly. Glasses were not normally tailored to a prescription, either. You basically just tried on pairs of varying strength until you found something that worked."
The restaurant proprietor tells Mugen that the yakuza are made men, a Mafia term that certainly didn't yet exist in Edo Japan, but it's appropriate, since it establishes that the guys are the period's equivalent of mob-style gangsters.
The two men knocked out by Oniwakamaru have scarves around their shoulders which are made up of the design of the American flag: one blue with white stars, the other with red and white stripes. No one in the Edo Period would be wearing scarves made to resemble the American flag.
No one actually says Jin's name in the whole episode.
Mugen fighting style
Possibly not an anachronism: many have noted the resemblance between Mugen's fighting style and the African-derived Brazilian martial art of capoeira. Capoeira has existed since the 1620s, so there's certainly a possibility that a practitioner of this fighting art, which originated as a form of self-defense for runaway slaves, might have crossed Mugen's path; and Mugen's ability to learn fast and thoroughly from one combat example is already well documented.
In contrast to 99.9% traditional Jin, Mugen is pretty much an original invention from top to bottom. The adaptation of the already loosely-cut gi and hakama to a hip-hop baggy style by just cutting the trousers short and untucking the jacket is simply inspired; I'm counting this as an anachronism because (a) this costume wasn't worn this way in the Edo period and (b) it's a conscious attempt to make period clothing resemble a modern style. I'm a little hesitant about including his metal-soled geta however; while there were a lot of different geta types popular in the Tokugawa era, I feel pretty safe in saying metal soles weren't one of them, so these are purely original, neither period nor modern in form. Ditto his sword, which we've pretty much agreed is one-of-a-kind.