Is there a way to get on-screen pop-up notifications for new e-mails in Mail on the iPad 2?
In addition to the Alert Style options which you mention setting, you also need to turn ON Notification Center for those alerts to appear in Notification Center, and also set View in Lock Screen to ON, to see them when your screen is locked.  

Otherwise you will only see them when you are using the phone at the time that they come in, and if you already have Mail open, then they won't appear at all.