I have an older iMac from 2011 and can't update my operating system to work with my printer.

I'm sorry but you have it backwards. One does not update an OS to conform to a printer. Instead, you need to update your printer's driver to conform to the Macs operating system. Check the printer's web site for a compatible download.

Running a Mac mini 2011 and High Sierra is the End of the Road for anything newer.

Highest OS a 2011 iMac can run is 10.13.6, but what printer is it?