Since Google disabled Exchange push capability on free Gmail, there is no way to push e-mail to IOS Mail app. Standard Gmail app does not provide selective notifications, like IOS Mail’s VIP feature (where there is a separate notification profile for selected contacts).
Enter myMail app.
It has notifications, it works with Gmail (and Gmail with comany domains using Google apps), it does push e-mail, and, most importantly, it has notifications that can be enabled per folder.
- In Gmail, visit Settings -> Filters, create new filter
- Put conditions to select important e-mail. Useful condition is has:circle in “has” field – this includes all people you follow in Google+
- Check “Apply a label” and select existing or create a new label
- In myMail settings, enable notifications (Notifications->Filter->Folders) per folder and select the label you just added