Is this a writing exercise, or a full-on PoD? If the latter, then, well, nothing happens. Rome at its height was the definition of “cosmopolitan”; there was no shortage of Egyptians, Syrians, Jews, Gauls, Lybians, and maybe even a handful of Germans and Persians, who didn’t fit into the Greco-Roman mainstream and would have interacted with their surroundings (mutatis mutandis) much as a foreigner in New York City would today. Is anybody shocked or offended when a Pakistani shopkeeper, a Nigerian expat, or a Chinese diplomat doesn’t take much interest in the city’s Thanksgiving celebrations?
If the former, you’ll need to specify a lot more about the “barbarian’s” origins and social setting. Is he a rich Syrian staying at his Roman friend’s opulent villa, or is he a manumitted Germanic slave’s son returning to his filthy insula full of resentment at disappointed prospects, or is he a Jewish traveller with secret Zealous sympathies whose faith has been renewed at the sight of Rome’s decadence and the thought of his oppressed brethren at home? Each one of those will have very different ways of moving through the streets.
EDIT: But if this post is subtly (maybe too subtly!) referencing Christianity, then... again, the answer is almost certainly going to be "nothing happens". Non-participation in a major religious event like Saturnalia may have constituted material for denunciation during the more intense phases of persecution, but (1) plenty of perfectly orthodox, even pious Romans avoided the wine-fuelled haze of Saturnalia to no consequence, and (2) there were much more significant signs, and consequences, of Christianity to a serious inquisitor. Remember, the main concern of the Persecutions was to deal with what, in the inquisitors' eyes, was basically a political crime rather than a religious one. A random barbarian quietly avoiding the Roman equivalent of Carnival is of a lot less concern than a fervent Bishop hosting meetings on nonviolent resistance in unmarked safe houses. At absolute worst for our hypothetical Christian, he gets denounced by some bitter busybody, brought up before a state inquisitor, and told to make a token sacrifice to the Emperor - which he likely does. If he doesn't, he meets the same fate as the rest of his coreligionists in the same situation. Most certainly nothing happens in regard to the development of Christianity, which occurred precisely because of events like this.
