How can I upgrade an old iMac?

To get a better identification that will better help us help you, find the serial number on the bottom of the stand's "foot":

Enter the serial number into this web page:

https://everymac.com/ultimate-mac-lookup/

The return will contain the important SUB-MODEL that uniquely shows your iMac's capabilities and OS limitations. The sub-model will contain a year and possibly a modifier: Early, Mid, or Late. Both the year and the modifier are important to this discussion.

I used to refer users to Apple's version of the same function but a recent page remodel removed the sub-model from the return, a great resource lost.

⚠️ NOTE: for security reasons, do NOT post a serial number here. The model will redact it and we will be back the Square One.

Unfortunately that model number has been used for eight different Macs. From 2012 to 2017. See if you can reinstall macOS from Recovery Mode.

How to reinstall macOS - Apple Support