how to update an old iMac 2013 to allow the use of Microsoft 365

You will need a Mac capable of running macOS Monterey (12.*) or newer. Microsoft only supports the current and two previous releases of macOS, and has retired all prior Office suites (and their license servers) except Office 2021 for Mac and the current Microsoft 365 subscription suite of applications.

The last operating system that a 2013 iMac can update to is macOS Catalina (10.15.*) and that is the oldest operating system that the current, free LibreOffice Suite 24.2.1 supports.

You can’t. Microsoft no longer offers a version of Office that will run on that machine.

You may still be able to use the web based Office apps, but if you want locally installed apps, you’ll need a newer Mac that can run a supported version of macOS.