The Spanish empujar (“to push”) has the same common ancestor as the English for the same, push: the Latin pulsare.
Pulsare meant, in Latin, “to beat”. A push is a sort of beat, in both senses: a punch and, a punch happening over and over again!
From the same root we also get the English, to pulse, of course. As does… impulse. Yes, an impulse is indeed a strong punch!
The sound here is a variation of the sh-to-j pattern, where variations of the s/x/sh/soft-g sound in Latin turned into the “j” in Spanish (via the Arabic influence) but remained the same as it transformed from Latin into educated English. Hence the “sh” sound in “push”!