Did Jews in the 1st century generally hold to the belief that humans are by nature immortal, or did they believe in some form of conditionalism (sorry if that's not the correct term, as I'm coming from a Christian background), whereby only the "righteous" live forever, etc.
The House of Hillel and the House of Shammai dispute about "the day of judgment," apparently after death (Rosh Hashana 16b). The House of Shammai say that the wicked in gehinnom are released after they cry out to God. The House of Hillel say that the wicked are no longer punished after 12 months, and are spread under the legs of the righteous. The continuation (which might be from the House of Hillel, or possibly a later anonymous addition) is that certain people are punished such that "hell is finished and they are still not finished."
Since Rabbi Yochanan ben Zakkai, who was of high standing at the time of the destruction of the Temple (70 CE), was "the least of Hillel's students" (Sukka 28a), I think this can safely be dated to the first century.
There certainly was the idea that Jewish souls are immortal, but there also were conditions, especially in the Temple Periods, but also after, though they were progressively less strict. These two concepts seem paradoxical, so let me explain. Since the original sin, the divine soul (which is separate from the animating soul present in all creations, progressively more evident from the inanimate to life forms and from there to the higher life forms) was split into many base souls, which can further be split into pieces, and those into pieces, ad infinitum.The goal of the split is for each piece (which is placed in a Jew) to collectively fulfill the goal of the base soul. Note: The following is a point of contention among the early commentators, but your question is a good one and I won't shy away from telling you one of the most prominent opinions. It is also the harshest.(Generally, when Jews approach a disagreement such as this, they accept every opinion as true in some way, just that the whole truth is not known to anyone. This is a disagreement in the philosophy of crime and punishment and therefore serves to make Jews be more steadfast in the service of G-d. This type of teaching, which is useful on the path to repentance, is known as mussar: Each piece can be destroyed totally, preventing the identity of that Jew from reaching the afterlife or being resurrected. They will simply be gone, not sent to purgatory.) However, according to all opinions, the base souls will remain, because the splits are more akin to smaller lamps lit from a central source than a fragmentation of substance; splitting doesn't detract from the central flame. Most of this is in or derived from Maimonides.