The obvious answer to this is yes, it was, because somebody did--maybe. Archaeologists today have a handful of tools, such as DNA testing (which traces ancestry) and isotope analysis (which documents the physical environment in which one grew up), that allow them to get a general idea of where a skeleton's owner comes from--if preservation permits. And with this there have been a couple of tentative identifications of people with East Asian ancestry in Roman burial contexts--the ones I am familiar with is a person buried in Vignari, in southern Italy, with maternal ancestry tracing to East Asia, and an individual buried near London who may have grown up in China (that case is pretty tentative, as the article details). I linked to news articles (albeit both written by serious scholars) because I have not seen in really detailed treatment taken up in an academic source because, end of the day, there is not much to say about it besides your probable reaction of "huh that's neat". Even if these are 100% confirmed, and actually from the territory controlled by the Han imperial court, they do not actually answer your questions.

We can say a bit more going the other direction--people from the territory of the Roman empire going to the territory of the Han empire. In 166 CE an "embassy" of the Emperor "Andun" (Marcus Aurelius Antoninus?) of "Daqin" (the Chinese term for Rome) arrived in Rinan (modern Vietnam) bearing tribute of tortoise shells, Elephant tusks and rhinoceros horns. The problem, you might note, is that those are the sorts of things that one might go to southeast Asia to get, not exactly something you would bring to the region from the Mediterranean. Most scholars believe these were quick thinking merchants engaging in cabotage in the area before being picked up by Han soldiers, and claimed they were imperial representatives. On one hand, this does indicate a certain level of possibility, that people from the Roman empire could end up in Vietnam, on the other hand the fact that we have this story indicates a certain unusualness of this experience. But we can say it was not common.

At the very least though, we do know something of the lines of connection across the Indian Ocean that somebody might travel on, either as a sightseer (a young aristocrat traveling to India is mentioned by Lucian), a merchant, or as property. In the western Indian Ocean this has been well studied, with a handful of very thorough excavations in ports on Egypt's Red Sea coast, along with somewhat more incidental research in the Arabian peninsula, western India, and east Africa as well as some literary remains, has illuminated a highly interconnected world of commerce and trade. The eastern Indian Ocean is rather less well understood--there is less archaeology in general and no region with preservation conditions like Egypt's--but there certainly has been found signs of contacts (such as the famed Indo-Pacific beads) and something happened in the early first millennium leading to a heavy cultural influence in southeast Asia from the subcontinent.

The land route, which dominates popular imagination of "the Silk Road" but almost certainly not reality, is even less understood. Central Asia was and is a very diverse, multicultural place where individuals might travel far in their life. The common fear of the Han court of peasants fleeing to the Steppe, and people captured by raids, allows one to construct a plausible fictional narrative of somebody from China ending up in Rome, but there is not much in the way of documented examples.

If you are curious about this whole topic I would recommend Raoul McLaughlin's Rome and the Distant East as a great readable introduction.