Install MacOS Monterey on portable hard drive and also use same HDD for time machine?

There is a much easier solution.

Use that external hard disk drive to back up that Mac with Time Machine as you proposed. Don't partition it and don't install macOS on it.

Then, you can completely erase the MBA's internal storage by following What to do before you sell, give away, or trade in your Mac - Apple Support.

In Step 7, choose the key combination that installs the latest macOS that is compatible with your Mac. That will be Monterey, as you stated.

When it finishes installing (which might take an hour or two, depending on your Internet connection speed and other factors beyond your control) the setup assistant will ask if you want to transfer information from another Mac or its Time Machine backup. Choose that option. That will also take a lot of time, probably even more time than installing macOS, but let it finish.

It has been my experience that the above procedure is the most practicable way to upgrade macOS on a Mac with limited storage. Consider it but you are welcome to consider all other options too. Personally I advocate having more than just one Time Machine backup as well.

Good place to start even if not sure where to place this question.

Firstly, Attempting to Install Monterey on an External Drive is very problematic at Best. That does not mean it is impossible but very iffy.

Secondly - Partitioning an External Drive to be used for Two Very efferent purposes - Bootable Drive and TM Backup is not so good.

Reason, all the Eggs in One Bucket and the Basket fails - Scrambled Eggs

Thirdly and most important in this computers' setup.

There are reports appearing on the ASC Forums regarding failed installation of Monterey on older Qualifying Apple Computers. In most cases this is related to having a NON Apple Original Internal Drive ?  

The EFI Boot ROM included in Monterey is verifying for Original Apple Drive and if not found , may Balk or fail to upgrade to Monterey. 

Reports indicate the EFI BOOT ROM is Required to Flash the Computer and is a  requirement for a Successful Monterey Installation.

Fourthly - Even if you were able to clear Enough Space on the 128 GB Drive , it is going to be a challenge to keep at least 15%  to 20% of the Drive Total Capacity as Empty Space for good operations of the computer and for additional updates to Monterey to download, expand and install.

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

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The final word from Apple on Managing the " Other/ System Data “ Category

Other / System Data: 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.

EDITED

John Galt wrote:

Installing Catalina will require more thought on my part.

The first thought that comes to mind would require installing that Mac's originally installed operating system, followed by creating a bootable Catalina installation USB drive: How to create a bootable installer for macOS - Apple Support.

Then, erase that Mac, boot from that USB drive, install Catalina, and when asked transfer contents from the Time Machine backup. It's up to you to decide whether it's worth the effort.

Be sure to create that TM backup before you do anything else. Preferably, two TM backups.

Only as a Point of Reference regarding Internal Drive Installation of Monterey on a 128 GB SSD Drive .

Regarding the 15% - 20 % Empty Space even after Wiping the Entire Drive and Installing a Virginal version of Monterey.

Arrive at your own conclusions

Don't use a USB flash drive for Time Machine. Its capacity is insufficient and it will probably be too slow anyway. A 2 TB hard disk drive will be more than adequate.

My suggested procedure is not applicable to installing anything other than the Mac's originally installed operating system or the latest compatible version. Installing Catalina will require more thought on my part. So will your need to use the Finder to move files to and from your iPhone.

Using iCloud Drive with the iOS Files app will work, but I understand if you are committed to using a different workflow.

Great! Thanks for your report of success.

I knew it would be a cumbersome workaround but I'm glad you persisted with it.