Is the scribing quest lasting for all time or will it go away? I could go finish it to get it.

Scott86

If it's temporary, I could finish it. I thought I may not scribe anything; I thought my abilities for nightblade could be good enough with archery.

NoticeMeArkay

The bow skillline isn't scribing exclusive. It only offers one additional spell for it on top of the already existing ones.
It does so with every weapon skill line and a few guild skill lines. Also Soul magic.

Scribing won't disappear after the crown store quest is gone. It will remain and you can pick it up whenever you feel like it now that it's part of the base game.