Problems with upgrade to monterey

Well there are definitely some major issues. I will list them in order and let you know what you need to do.

1. First there is no backup for this computer. This is EXTREMELY shortsighted and is guarantee you will lose all of your data. It is not a matter of "if" it is a matter of "when". Apple has a backup app pre-installed on your computer called Time Machine which only requires an external HD to back up to. You can learn about TM in Back up your Mac with Time Machine. That is your #1 priority, get your computer backed up and let TM backup as it is designed to. If you need a good quality but reasonably priced external HD I recommend the OWC Mercury Elite Pro , get one that is at least 2 TB. Please buy that today and when it arrives begin using TM!
2. Next, Mac OS has been damaged somehow. I can tell as their is no security software installed. This is easily fixed. Restart in the Recovery Partition by holding down Command + R on startup. When it starts, you will see a selection for installing Mac OS, do that. But DO NOT do it before backing up. While this type of install is safe you need to backup to safeguard your data. The install will not overwrite any of your data, what it will do is to fill in the missing pieces. After that please restart in Safe Mode and then restart normally. I would then recommend that you download the free version of Malwarebytes and run it to check to see if you have any malware installed.
3. Finally, attempt to install Monterey.

Please restart the computer in Safe Mode per the directions in the  How to use safe mode on your Mac - Apple Support link. 

Note: If you are using a wireless keyboard please either plug it in using the supplied Lightning cable that shipped with it. If you are using a legacy keyboard or a third-party keyboard that does not have re-charging abilities, please use any USB keyboard. Also note that a Safe Mode restart may require you to hold down the shift key for up to 5 minutes before the computer starts in Safe Mode. You will know it is in Safe Mode when you see “Safe Mode” in the upper right of the display.

After restarting in Safe Mode, then open System Preferences - Software Update and update the computer. If that does not work, restart normally and do the following:

It will help us get a good idea of what is causing your problem if you provide a report of your system, that way we do not have to play 20 questions with you. The report we are requesting does NOT provide any personal information and is extremely safe to use. Please navigate to www.Etrecheck.com and download the free version of EtreCheck. Once you have you downloaded the app and installed it, please run the report and save it. This report will help us get a good idea what has been installed on your system and help us be able to diagnose what may be wrong.

When you have your report, you can attach it when you reply to this message and we can then review it and help you determine what is needed to get your system running well again. 

For instructions on how to download your EtreCheck report and attach it to your reply to this message please click How to use the Add Text Feature When Post… - Apple Community

You can try to download monterey manually via this link How to get old versions of macOS - Apple Support

Got it and good luck!!

Your welcome.