if you are talking about gmail in default mail-app, you can get push by creating an extra icloud-account and forward your gmail to that. See first option here: https://www.technorms.com/45016/gmail-push-notifications-iphone
Not mentioned (as far as I could see, after a quick look), you should change outgoing server to google server. So when you reply it will be from your gmail.
There is no such a setting controlling how often you can check your app update in iOS unfortunately.
However, judging by the update frequency of various apps, I doubt more frequent check would bring any difference. Also, too frequent update check, just like checking new email, will have negative impact on battery life.
Is there a specific reason driving you to think checking the app update more frequently is good?