I was/am in the same situation with a 2012 Mac Mini running Catalina. You can get a VMware Fusion license for free for personal use and use that to install Big Sur or Monterey. You will need to setup an account which is the gateway to receive the license and download.
You can install the MacOS as a clean install by getting the system installer of your choice using this page for example:
If you want to use your existing Catalina volume to keep everything intact, then you need to make a disk image of that partition and VMware Fusion can use that to install what you currently have now, and then you update from there.
It works OK but is not going to be as fast as a regular system volume but it does work better running from an SSD so if you have that, I would use it but you can try it first on a regular drive. It’s not hard to do but there is a bit of a learning curve if you never setup a VM. Also, if you have a Windows installation you can use Fusion for that too.