iMac (21.5-inch, Late 2013) How can I update it

You will need to restore the device using a computer. The process will erase the device. Later you can restore your data from a recent back-up. Instructions here…

If you forgot your iPhone passcode - Apple Support

As for your question about your iMac, you haven't told us what version software you are trying to update to. In all likelihood, an iMac that old won't be able to update to any newer software.

Regarding your Mac question, here is the MacTracker information for your model:

Note that it can be updated to iOS 10.15.7 (Catalina), however it is tagged as “obsolete” and out of hardware support.

BTW, you can download the MacTracker app from the Mac App Store.