As originally published in 1982's "New Teen Titans" #16 preview (and the first Zoo Crew issue), Starro was attempting to de-evolve denizens of Earth-C into their realistic-acting animal ancestors; however it also affected Earth-1's humans (who started behaving simian-like). Superman was prompted to investigate, and wound up being hurled by Starro's de-evolving ray into Earth-C (along with meteor fragments that gave the Crew members their powers). After Supes and the Crew head to the Earth-C universe's Pluto where Starro was based, Starro reveals that several years ago, Aquaman blew him to bits in some battle (apparently in Adventure Comics' Aquaman run of stories); one of the pieces of Starro drifted through some sort of black hole into Earth-C's universe, where it regenerated into the Starro we see here. The Zoo Crew defeat Starro, and he's not seen again.... until the recent "Final Ark" miniseries where Starro plays a big role.
In said miniseries, Cap and the gang still recognize Starro from their first fight with him, and even still recall meeting Superman... so guessing the Zoo Crew's origin (save presumably no references to "Earth-1" and no Gold Kryptonite wielded by Starro) as first published still stands (no post-Crisis version of their origin). While it's possible post-Crisis Starro on Earth-C could simply be an original, Earth-C-universe counterpart of the DCU Starro (with his own backstory, etc.), it seems easier to assume (for the Zoo Crew and a pile of other stories dating back to, well, "Brave and the Bold" #28) that Starro wasn't rebooted.
From what I recall, that miniseries also claimed that only one method of time-travel could be used *once* by a time-traveller. STupid beyond belief (especially considering, well, all those Barry Allen time-travel stories still happened and, uh, sort of undermining the whole premise of "Rip Hunter, Time Master" itself). Think this miniseries (or its time-travel premise at least) got tossed quickly... by the 90s, Hunter was IIRC working with the Linear Men, the "timecops" who appeared in some DC titles (particularly Superman) of the time.Rip Hunter, Time Master got a big Post-Crisis reboot in the 8-part Time Masters miniseries published around 1990. I have heard that the rebooted version has been retconned and replaced by a different version, but I could have some of this wrong. Aside from reading that miniseries, I am no great expert on Rip Hunter's continuity—and time travelers in general are prone to run into alternate versions of themselves, etc. (Think of Marvel's "Kang," for instance.)