How to upgrade imac 13(MacOS Catalina version 10.15.7)

Whether "iMac13" means "a 2013 iMac" or "an iMac whose model ID begins with 'iMac13,'", the answer would be the same. There are no upgrades available to anything higher than Catalina.

(MacTracker indicates that model IDs beginning with "iMac13," correspond to iMacs from Late 2012 and Early 2013.)

Catalina is still good enough to run current versions of

• The Firefox web browser
• The LibreOffice office suite
• The GNU Image Manipulation Program (GIMP)
• The commercial Affinity V2 applications (Photo, Designer, Publisher)
• The backup utilities CCC v6 (not current, but still supported) and SuperDuper!

It is no longer supported by vendors like Microsoft and Adobe. So while there are some things the OP could do to stretch the life of the machine if he or she does "not want to buy a new hardware", the OP may eventually need to "bite the bullet" and do just that.

If you have a 2012 or 2013 iMac..?

see > Identify your iMac model - Apple Support

Then 10.15.7 Catalina is the highest macOS that it can run.

... and, as mentioned by the others, this iMac model cannot be upgraded past macOS Catalina 10.15.7. If you require a newer version of macOS, you will need to get a newer Mac.

If cost is an issue, Apple does sell refurbished Macs, as well as, other third-party sources, like OWC.

Without knowing the model year of the computer we cannot answer the question. If you do not know, click the Apple symbol in the upper left of the display and then click About This Mac and the window that opens will provide the model year.