Not sure about the primary schools but girls secondary schools in Bishopstown are Mount Mercy, cork city centre has St Aloysius, Douglas has Regina Mundi , all these are well regarded girls schools within commuting distance from bishopstown - cork quite small really . Also if you move west of the city to Ballincollig there's Colaiste Coilm ( excellent reputation for music , science and irish) . None of these are private - there are a couple of private girls schools as well - st angelas , scoil mhuire both on patricks hill.