The answer is ‘yes’. Most traditional church history centres on the Roman Empire, but we know that in the earliest periods Christianity spread in the Syriac-language sphere and developed into the Church of the East throughout the Parthian and then Sassanian empires, and beyond into India and China.
In the case of India, the tradition contained in the Acts of Thomas suggests that Thomas went to India. While difficult to put much historical reliance upon it, the text itself suggests that the tradition that Thomas went to India dates at least to the 3rd century. To be fair, it is not impossible that Thomas went to India, just very difficult to verify. The document discusses a King Gundaphar, who appears to be a verified figure from the 1st century. Anyway, whether Thomas got there or not is pretty difficult to establish. A second mission is recorded, in that Pantaenus, a scholar from Alexandria, was sent from Alexandria around 179/180 AD. The account recorded in Eusebius suggests both that Christians were already there, and they had the Gospel according to Matthew in their language, thanks to the apostle Bartholomew.
Anyway, the Syrian church is very well attested throughout the Late Antique period, and records missionaries travelling from Edessa eastward, to the borders of Mesopotamia and beyond. On ehas to wait until the later fourth century for a number of references in sources to Christians in India, and from the early 5th century you begin to get details about Indian Christians being organised at a hierarchical level into the Persian church structure.
As for China, the earliest attestation is the Alopen monument, dated 781, which refers to a Alopen, a missionary, coming in 635 AD from Syria and bringing the scriptures with him. This coincided with a period of religious tolerance under T’ai-tsung which saw an early flourishing of Christianity. A 10th century Arab document records that Christianity had been all but eliminated in China. It's very likely that some individual Christians (traders, officials) had entered China before Alopen though.
Those are the short answers. S.H. Moffett’s “A History of Christianity in Asia” is a good starting place for this kind of thing.
Speaking specifically of Okinawa and Japan, there is no evidence. That means it's not impossible, but it's pretty unlikely.
I can tell you more about Okinawa, but I'll need access to my references to really give a good and detailed answer. A short answer would be to say that the Ryukyu Kingdom came into first contact with Christians during the European Age of Exploration, and quickly got a reputation for being a tiny nation that is unusually and incredibly hospitable to shipwrecked sailors. This comes from the importance of maritime customs on their own culture.
There weren't many shipwrecks there, but there were enough to bring the Ryukyu Kingdom an outsize amount of knowledge and respect amongst seafaring Europeans--enough that in the 19th century, there was enough money raised to send a Christian missionary named Bettelheim, who quickly alienated the people he came to convert through his ridiculous antics. That didn't stop him from continuing to try, and he served as a really terrible interpreter when Perry rolled in and forced an unequal treaty with the kingdom as a practice run for Japan.
And that's really the first extensive Christian-focused contact with Okinawa.
Of course, Christianity came in force to Japan much earlier, with Francis Xavier in the 16th century. And, as you likely know, got quickly stamped out and went underground.
Vietnam, according to available evidence, encountered Christian missionaries only in the 1500s. I suspect some contact may have occurred earlier, if only for the active commercial networks that depended upon Vietnam's ports. Also, if I'm not mistaken, Vietnam only reached from the north to the mid-point of where it is today at that time. So, the Champa Kingdom or Khmer Empire may have records of contact. Anything earlier and it's doubtful the records exist outside classical Chinese histories. Wish I could be of more help.