My Instagram notifications don't work

Hello there and thank you for posting your question here at Apple Support Communities! From the included screenshot I see that you aren't in the right preference pane. To configure notifications for a specific app go to Settings > Notifications > (Application that you want to configure). More info on how to configure notification settings for applications can be found here. Hope this helps and have a nice day.

As indicated: turn off low power mode, and then turn on Background App Refresh.