I think you are referring to this drop down menu at the top right corner in the admin portal. In short - it is not possible to view Ann only ext there. You have several good options
1- a user from said department who has Admin access can edit the ext. by going to Phone System > Groups > Ann only ext... (or at least Phone System admin)
2 - A standard user can log in the admin portal using the following credentials instead of email + pwd
- Company Main number
- Ext number of the Ann only ext
- password
3 - Sounds a bit weird to constantly be changing the Ann only ext. It might be possible to create several Ann only ext. and set up different call handling rules for them. Set it and forget it. If you explain a bit more about the way you are using the Ann only rules, I could try to recommend an alternative. Of course, it might be the case that having someone edit the Ann only ext is the best way for you.