Was the issue of bastards handled very differently prior to Christianity? How did the West compare to the places like China or Japan in the middle ages on the issue of bastards and their place in society?
In ancient Japan things were a bit different. When you read books like The Tale of Genji, all the nobility have separate bedrooms from their spouse, and marriage is for producing heirs(and even then parenthood is iffy). It definitely seems as though, so long as a first child is produced with the appropriate partner, people from that class were free to visit pretty much whoever they liked.
Is it even possible to be a bastard in a society where monogamy isn't the norm?
It was stigmatized to a very varying extent, given place and time. You're probably thinking of "Yer a bastard, John Snow" and all that, but the reality was that bastard sons were often a kind of resource for noblemen and royalty. The French kings, for example, often had bastards, and those that showed promise could be given positions of power or titles. They were often more loyal than other nobility, since if they owed everything to the king. Piss him off, and you're kicked out on your ass.
Particularly notable bastards could found houses of their own. The prefix fitz- was used from the 1600s on to denote bastard houses. Fitzjames for example ( http://en.wikipedia.org/wiki/Fitzjames). Fitzroy, obviously, means bastard son of the king. This doesn't mean all fitz- houses come from illegitimate children, though. It's an old Norman word.
Louis XIV had tons of illegitimate children. ( http://en.wikipedia.org/w/index.php?title=Louis_XIV_of_France#Issue) Some were legitimized, some were not. Of those that were legitimized, they tended to do quite well. Look through the reference. They were named 'de Bourbon' instead of 'de France' as the result of their birth.
But yeah, it basically came down to matters of inheritance being the biggest issue.
In Japan, they had a different outlook on it than in the west. The social status of the woman was somewhat irrelevant - children drew status from their father. So when the 4th shogun knocked up the daughter of a grocer, the resulting son was a valid heir, who did in fact become the 5th shogun.
How does the concept of illegitimate children correlate to Christianity anyway?