Microsoft Outlook.com app notification
You can't have email push notifications over POP3. It's an old protocol and simply doesn't support push. It doesn't matter what client you use or what kind of mail server.

If you can set up Outlook to access Gmail through IMAP instead of POP3, and if Outlook supports IMAP IDLE for push notifications, then you'll get push notifications. Otherwise, you might have to use a mail client that does support IMAP IDLE, such as K-9 Mail.