Microsoft 365 can run on macOS Ventura 13 only by using an older Office version.
Microsoft ended support for Ventura with Office version 16.101, released in September 2025. Newer Microsoft 365 installations and updates require macOS Sonoma 14 or later.
Uninstall your current Ms365, then downgrade to version 16.101 by using this installer.
https://officecdn.microsoft.com/pr/C1297A47-86C4-4C1F-97FA-950631F94777/MacAutoupdate/Microsoft_365_and_Office_16.101.25091314_Installer.pkg?apid=ets
After the install,
You can use the instructions below to get rid of the auto updates and any reminder to ask you to update, since you cannot really update any further on your Office apps on this macOS.
Close all Office apps.
Open Finder, then press Command + Shift + G.
Paste this path and press Enter:
/Library/Application Support/Microsoft/MAU2.0/
Find Microsoft AutoUpdate.app and rename it to Microsoft AutoUpdate.app.disabled
Enter your Mac administrator password if prompted, then restart your Mac.