I think this is where the confusion lies; most other spells say the wizard can make a shooting attack, but Push says the target 'suffers a +10 attack'. Does that mean he already counts as being hit?
No, it means you roll and add +10 and then compare to the opponent's roll + combat modifier, just like any other attack.
It's not called a shooting attack because it doesn't use the shooting attack modifiers table but it does use the basic attack rolling structure.
If you win then minus the opponent's armour score from your total attack roll, the result is the number of inches pushed.