You can manage push alerts on a per account basis from the in-app settings page, but keep in mind that the new notification options rather surprisingly work only with the new inbox.
Speaking of which, the revamped Gmail inbox lets you file messages into the following inbox categories – Primary, Social, Promotions, Updates and Forums.
What’s New in Version 2.3.14159
– New notifications options: Get notified for all messages, only important messages, or none at all, on a per account basis. Customize these settings from the in-app settings page.
– Meet the new inbox: If you’ve enabled the new inbox, your mail is now grouped into categories so that you can see what’s new at a glance and decide which emails you want to read when. Primary mail is shown first and it’s easy to access and keep track of email in other categories by opening the menu. Configure this feature by going to Gmail.com on your computer.
I know you are so excited right now, so right now I want you to head forward to the App Store and update your Gmail’s app to enjoy using all of these great features…