A case can very much be made that Job predates Sinai, and that, in his day, priesthood was assumed by the father of the house, who, as the head of the house, functioned as the priest for his family. This no doubt was passed on from Abel and his children to Job's day as the norm (even though it's not certain the nature of the sacrifice Cain and Abel offer in Genesis 4 in particular); after all, Abel was not a Levitical priest, and of course well predates Sinai.
In the book of Job we find Job tentatively offering sacrifice for sin "in case [his children] have sinned and have scorned God in their hearts"—"sanctifying them" with "burnt offerings" (1:5); specifically, he offered this after each of their birthday feasts at their respective houses (1:4), in case, in the heat of the feasting, his children had committed sins (most likely the 'debauchery' to which excessive partying can lend itself all too readily). Notice that the sacrifice was (1) for sins, (2) to sanctify them (or make them holy [again]). This constitutes sacrifice for sins before Sinai.
As noted by Nigel, however, Christians have traditionally always seen the killing of the animal to cover Adam and Eve's nakedness to be an instance of sacrifice for sin, where the cost of covering the shame of Adam and Eve is the life of an animal, but which God paradoxically accepts—and in this instance, freely offers. If this doesn't have a spiritual overtone in Genesis, it most certainly does in Revelation 3:18-19 (DRB):
I counsel thee to buy of me gold fire tried, that thou mayest be made rich; and mayest be clothed in white garments, and that the shame of thy nakedness may not appear; and anoint thy eyes with eyesalve, that thou mayest see. Such as I love, I rebuke and chastise. Be zealous therefore, and [repent].