You can walk into any bank in Italy to open an account. I did this in Bolzano with Cassa di Risparmio(Sparkasse), Monte dei Paschi,
Raiffeisenbank and Volksbank.
Volksbank has a nice service.They have ready made VISA debit cards without the clients name on which will be activated after the account is open.
Better you call them before to arrange an appointment and ask for their requirements.