Since a positronium has no proton, I would suggest $Z = 0$ but I'm not sure if you can see the electron as the center of mass and therefore $Z = 1$.
The atomic number is the number of protons. Positronium has no protons, so it definitely cannot have a non-zero atomic number.
But positronium is also not an atom and so it cannot even be said to have atomic number number. It's simply a bound state of an electron and it's anti-particle.
The positronium is technically not an element, it can only be said to be a particle. This question may not be so physics-related, so please go to the chemistry site and check.