I don't think there's any way to access iOS notifications outside the device. If the notifications you care about are also available via email you can disable t

I don't think there's any way to access iOS notifications outside the device. If the notifications you care about are also available via email you can disable the official app's notifications and use email rules to redirect specific email notifications to something like Pushover, essentially rolling your own push notifications where you can use rules to control what gets through.