The Planar Bridge stole a Niv-Mizzet from my library and had it in play.
On my turn i began casting multiple spells, the opponent’s Niv-Mizzet triggered on casting my casting spells.
(so he drew cards, and i took damage)
That is correct. Niv-Mizzet Parun’s ability reads “When any player casts a spell, Draw a card” (emphasis mine)