How did Zoroastrianism influence the way Persian empire was governed?

It didn't. The Empire was governed according to simple rules - local government was by local people according to their traditional model. This was supervised by 20 provincial governors who were also responsible for security and tax collection. A king and his council oversighted the governors.

Zoroaster was a Persian god who had a cult following; the king accepted him an important god. The peoples of the Empire retained their customary gods.