First off, there are clearly some background concepts from the Hebrew Bible that had a big influence on later theology here, both Jewish and Christian, on humanity and sinfulness.

Probably the most important general concept to bear in mind relates to the fact that Adam, of Adam and Eve fame (of the Garden of Eden story in Genesis 2-3), is a Hebrew word that simply means "man" or "human," and that this Adam is clearly presented in Genesis as an individualistic representative of humanity itself.

More on that in a second; but there are several other things that might be relevant here, too. For example, Genesis 6:5 and 8:21 say that "inclination of [the thoughts of] adam's heart is bad/evil his entire life"—an inclination known in later Jewish theology as the yetzer hara, the "evil inclination." Further, at several other points in the Hebrew Bible, the universality of sinfulness is emphasized: "for there is no one who does not sin" (1 Kings 8:46; see also Ecclesiastes 7:20). But, other than the fact that Genesis 6:5 and 8:21 use the generic term adam, in neither of these instances is this associated particularly with the individual Adam from Genesis 2-3. (In fact, the observation in Genesis 6:5 and 8:21 is associated with the later events that prompt God to send the flood.)

Of course, the most significant aspect of the transgression of Adam and Eve from Genesis 2-3 is its association with the origin and death and toil for humanity. It's here where we can most clearly see the kind of oscillation between the individual Adam and a collective adam. This is expressed poignantly in the early rabbinic Jewish commentary/midrash Sifra (though this itself isn't pre-Christian):

[Adam] was given only one commandment, a prohibition, which he transgressed, and see how many deaths Adam was condemned (to suffer as a result): his own, and those of all his descendants and his descendants' descendants to the end of all generations...

In terms of a focus on the original transgression itself, and its broader effects (in terms of the emergence of sin and death in the world), we can find clearer hints of this elsewhere in the Bible, for example in the deuterocanonical book of the Wisdom of Ben Sira, a.k.a. Sirach—though this places the blame squarely on Eve herself: "From a woman sin had its beginning, and because of her [=probably from her actions] we all die" (25:24). So here we see that the primeval transgression is not just about the introduction of death into the world, but that it's also explicitly identified as the origin of sin, too.

Now, another important aspect here pertains to the nature of the "death" implied in the Garden of . Although with the hindsight of modern academic analysis, it's pretty clear that Genesis 2-3 was indeed intended to be a story of the origins of literal human mortality (and toil), in ancient Jewish and Christian interpretation it wasn't associated specifically (or only) with literal mortality, but also a sort of "spiritual" death, too. For example, the early first century Jewish philosopher and Biblical exegete Philo of Alexandria, in his interpretation of Genesis 2:17, writes

[God] says, “On the day in which you shall eat of [the tree of knowledge], you shall die death [θανάτῳ ἀποθανεῖσθε].” And yet, though they do eat of the tree, not only do they not die, but they even beget children and become the causes of life for others. What, then, should we say? Namely, that death is of two kinds; the one being the death of the man, the other the particular death of the soul. Now, the death of the man is the separation of the soul from the body, while the death of the soul is the destruction of virtue and the entrance of wickedness [ὁ δὲ ψυχῆς θάνατος ἀρετῆς μὲν φθορά ἐστι, κακίας δὲ ἀνάληψις]. For which reason, God says that they will not merely “die [ἀποθανεῖν]” but that they will “die death [θανάτῳ ἀποθανεῖν],” clearly indicating that he is discussing not the common variety of death but the peculiar and extraordinary death, that is of the soul, entombed in passions and all sorts of wickedness.

(Of course, in its original context, the duplication/redundancy "die dying"—or "die death" in this translation—is simply a common Semitic literary idiom, and if it means anything different at all, it's probably just for emphasis, similar to things like "king of kings" or "song of songs"; and so here in Genesis 2:17 it would probably just be to emphasize the certainty of Adam's death. Indeed, most modern Biblical translations render Genesis 2:17 as "you will surely die.")

Now, there are some other Jewish sources from roughly around the time of the composition of the middle/later books of the New Testament (late first century CE) that also refer to the effects of Adam's sin in interesting terms. One problem here, though, is that it's uncertain to what extent the traditions behind these predate Christianity, or if there may have even been Christian influence on these. I've quoted a few of these texts in my comment here, but just to highlight one of these, in the apocryphal book 4 Ezra, we read

the first Adam, burdened with an evil heart, transgressed and was overcome, as were also all who were descended from him. Thus the disease became permanent; the Torah was in the people's heart along with the evil root, but what was good departed, and the evil remained. (3:21-22)

(It's highly probable that the "evil root" here is to be directly connected with traditions of the "evil inclination" that I referred to earlier.)

When we turn toward early Christianity itself, probably the most significant passage in the New Testament that influenced Christian theology on original sin is found in the fifth chapter of Paul's epistle to the Romans—in particular Romans 5:12: "sin came into the world through one man, and death came through sin, and so death spread to all because all have sinned."

Here we clearly have a similar echo to what we found in Sirach: recall "From a woman sin had its beginning, and because of her we all die." But perhaps most importantly, this passage became famous—or rather infamous—due to an early Latin translation (which appeared in the Vulgate, the "official" Catholic translation) in which the last words of this verse were rendered not "death spread to all because all have sinned," rather but "death spread to all in the one in whom all sinned"—cryptically suggesting that all of later humanity was somewhat mysteriously present "in" Adam. This would have a profound influence on Catholic (and other) theology on this original sin, in terms of the idea that humans not only inherited the effects of original sin (mortality, etc.), but were already born with something of the nature of personal responsibility for this sinfulness: a sinfulness for which one could justly be sentenced eternal punishment (even if one hadn't yet actually or consciously sinned in one's earthly life).

(More generally speaking on Romans 5, Menahem Kister has an interesting if speculative article that looks at Romans 5:12 and the surrounding verses here in light of some early Jewish/rabbinic traditions, like those in the Sifra that I referred to earlier: see his "Romans 5:12-21 against the Background of Torah-Theology and Hebrew Usage.")

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So, to summarize in the most general way possible, there were both some significant continuities and some discontinuities with the way that early Christian traditions on Adam, the original sin, and the emergence of death (and other associated effects) emerged and developed.