Generally speaking, I think the answer to your specific question of obtaining a Bible is "likelihood = 0". And the reason is pretty simple. There were no used book shops, no Amazon, no B&N, no Gideons, no etc. In other words, no way to even obtain a Bible.
Some things to consider:
Books in the 6th century were absolutely precious artifacts. Secular books were copied by hand by slaves trained as copyists. If you wanted a copy of Juvenal or Plato or something, you borrowed a book and took it to a copyist and haggled for his slaves' time and skill, quality of writing, etc. Either that, or you hired the slave and brought him to the book.
Bibles, of course were different. These were probably copied without thought of monetary cost --- they would have been copied as a spiritual work of love. By the sixth century, there are probably within the Church, copyists doing this work.
Bibles are also HUGE. The Codex Sinaiticus (oldest complete Bible MS) is 13x15 inches, made of vellum requiring the hides of more than 350 animals. It's said the cost in labour and materials would equal the wages of a single workman's entire career. This is not something a teenage boy will just happen across (unless he discovers a crime scene where a wandering monk has been robbed and the precious Bible he was carrying has been discarded as junk by the thieves).
The likelihood of a random teenager living in a tribe along the silk road even being literate enough to read a Bible in Greek or Syriac, to say nothing of Latin or Coptic, is about zero. The likelihood of there being a physical Bible for him to spend time with during that time frame is also about zero.
Much more likely scenario, especially given the time frame: a wandering priest or monk (who may or may not have some parts of a Bible with him) is travelling the Silk Road among the traders and your teenage boy hears him teach. He's attracted by the Truth and all the rest follows from there. Though, of course, he won't have to travel to Byzantium to learn about salvation! That he could learn directly from the monk, or even from the Gospel itself.
560s might be a little early for much Christian activity in the region. However, the Religion of Light (Christianity) was certainly established and flourishing in China by the 8th century, as one stele of the late 8th century there records 150 years of Christian activity in China (meaning activity since about 630s). So Christian activity along the Silk Road in the 560s might not be too far off, after all.
Interesting premise for a story, but I think you might have to scrap the idea of the hero getting a hold of an actual Bible.