In 2267, the Botany Bay was discovered by the USS Enterprise, where Rodriguez was among the 72 surviving members of Khan's original group of 84. During Khan's takeover the Enterprise, he was assigned to the armory. He was later imprisoned on the planet Ceti Alpha V. (TOS: "Space Seed") 6 months later 6 months later however Ceti Alpha XI exploded and laid waste to Ceti Alpha V, leaving only the transported cargo crates as shelter.
Whether he survived to serve Khan on the stolen USS Reliant 15 years later or was killed by the life forms of the planet is unknown. What ever the case, Rodriguez would be deceased. Star Trek II: The Wrath of Khan