If apps show up as being available for updates they should be updated. In regard to shutting notifications off you can shut them off totally or individually. SETTINGS/ NOTIFICATIONS.
If your talking about the icon that looks like this (see attached) then I don't think you can get rid of the red icon in the corner , there may be a Cydia tweak but not 100%?
If your talking about the alerts that pop up on the home screen you can shut them off like I explained above. If you talking about the App Store alert icon, like the writer above explained it can't be deleted.
If your talking about the icon that looks like this (see attached) then I don't think you can get rid of the red icon in the corner , there may be a Cydia tweak but not 100%?