Hide irritating window without admin?

Yes,
if you need access to iCloud, and you need to connect, then you need to know the password for that AppleID account.
I don't think you need to be logged in to an admin account just to clear the message, but it probably needs to be fixed from an admin account that has access to that AppleID account.
Sometimes the iCloud account login, along with its authentication, may become corrupted. iCloud blocks access until the information is corrected. Sometimes the account has to be deleted, then re-added, as you have seen.
A non-admin (who does not have the security responses, along with password) isn't likely to fix this, which I would expect will be the response from Apple when you call them, and you don't know the security setup for the account.
You (or whoever has set up the iCloud account) may need to call AppleID support to fix this.
(TDLR: you need the iCloud login info, or you have to fix with del and re-add, which is likely. Either way, you need AppleID account password)

 

Yes
But, the person who knows that AppleID password would need to take care of that for you.

Who is the admin for the "administrators_account_I_dont_have_access_to_right_now@hotmail.com"?
By the way, that message is NOT asking for the admin account password, but the password for authentication of the AppleID account that is listed for your Mac.
The admin account password, and the AppleID password are not the same.
You need the AppleID password - and, whoever is the admin for your Mac should know what that is, and should be able to help you.

Bottom line - you need to talk to whoever belongs to the AppleID.
There is no other way.