Can I update to Monterey using an installer on an external hard drive

It appears, from the above statements, this computer may have the tiny 128 GB SSD ? If this is correct - one is going to be hard pressed to create enough Empty Space to allow the Installer to Expand and then install.

Even if one is successful in make the Required Space - Empty Space afterwards is suggested to keep at least 15% of the Drive Total Capacity as Empty Space for good operations of the computer and for additional updates to Monterey . That 15% of 128 GB SSD equal 19.1 GB as Empty Space at all times.

What is “Other” storage on a Mac, and how can I clean it out?

Free up storage space on your Mac

OmniDiskSweeper Safe to use

How to delete Time Machine snapshots on your Mac

See used and available storage space on your Mac

The final word from Apple on Managing the " Other " Category

• Other: Contains files that don’t fall into the categories listed here. This category primarily includes files and data used by the system, such as log files, caches, VM files, and other runtime system resources. Also included are temporary files, fonts, app support files, and plug-ins. You can't manage the contents of this category. The contents are managed by macOS, and the category varies in size depending on the current state of your Mac.

Apple says you need:

If upgrading from macOS Sierra or later, your Mac needs 26GB of available storage to upgrade. If upgrading from an earlier release, your Mac needs up to 44GB of available storage.

I would try to get a little more free disk space.

Yes you can do that.

How much free disk space do you have w/o the installer on the Mac?