How do I stop getting notifications while cellular data is shut off for specific apps?

Howdy _Talldude,

Welcome to Apple Support Communities.

If I understand correctly, you want to know how to disable notifications from specific apps on your iPhone 5s. The article below provides a lot of great information about notifications and it will walk you through the process of disabling notifications for apps.

About Notifications on iPhone, iPad, and iPod touch - Apple Support

Turn off and change Notifications settings

If you have an app that could potentially send frequent notifications (like Mail with high-traffic email accounts or Twitter apps), your iOS device could wake frequently to display the notification, and affect battery life. You can disable Show on Lock screen for a particular app in Settings > Notifications, then choose the app and drag to the Do Not Include section.

You can also customize what you see in your Today view. Go to Settings > Notifications, and select which apps you would like to include. You can sort your notifications manually or by time. To sort manually, tap Edit in the upper right corner, select the apps you want, and drag them using the icon. You can drag the apps to the Include or Do Not Include list, and put the apps in your desired order.

Cheers.

Thank you for using the Apple Support Communities. Due to the nature of the notification issue, you may find more information by using another one of the Apple support resources provided here: https://getsupport.apple.com/GetproductgroupList.action.