The most obvious answer would be via the Silk Road; there was a spur from the east end of the Silk Road in Xian that went through Manchuria and the Korean peninsula to Japan, ending in Nara.
Quite so, but I mean the particulars of its journey, the lives of the people involved, etc.
Something like a Michener novel.
There was also a lot of seaborne trade. Some of it apparently got as far as Japan, through multiple intermediaries, no doubt.