His claim is: all beings that are alive also age. The counter-example makes so that his argument is false (hydras are alive, but do not age).
I didn't claim that being alive means you outlive the universe. Because that would be false. Everything alive is subject to the laws of thermodynamics (need energy to keep up the entropy-fighting machinery)--no food, no hydra.
The point is that a hydra that does not age is clearly violating the laws of thermodynamics; hence, eventually, they will age (as you pointed out).