Can I use Word 2011 with Monterey?

No. It's a 32-bit application and won't work in Catalina or newer.

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You can only use Microsoft Office365 (subscription), or Office2021 which is a one-time purchase and a perpetual license.

Your 2011 package is 32-bit. MacOS has required 64-bit applications for the past few years I believe.

This is the subscription version of Office365

https://www.microsoft.com/en-us/microsoft-365/buy/compare-all-microsoft-365-products?tab=1

This is the one-time license to buy a perpetual license (Office2021)

https://www.microsoft.com/en-us/store/cart?country=US&culture=en-us&rtc=1

You can still pass files back to your MBA, the file formats are compatible just not the app. Also, Word 2011 NEVER worked on Monterey. Since Big Sur all apps needed to be 64bit apps, Word 2011 as has been stated in this thread was a 32 bit app.

Word 2011 worked just fine on Monterey until the 12.2.1 Update, when it gave the "needs to be updated" notice and refused to load. I suppose I'll buy the 1-machine perpetual license. Sad that I won't be able to pass files back and forth with my old 2011MacBook Air anymore...

There is a deal going on where you could by a lifetime license of the most recent version of Microsoft Office which will be compatible for pretty much anything Apple releases for a while. It's only $49. No subscription.

https://shop.appleinsider.com/sales/microsoft-office-home-business-for-mac-2021-lifetime-license-2?utm_source=appleinsider.com&utm_medium=referral&utm_campaign=microsoft-office-home-business-for-mac-2021-lifetime-license-2_021622&utm_term=scsf-531779&utm_content=a0x1P000004UwEzQAK&scsonar=1

No. Word 2011 will not run with Monterey.

There was a 64-bit version update of Office 2016 for Mac but I don't know if that can be installed on Monterey. See: https://docs.microsoft.com/en-us/office/troubleshoot/office-for-mac/office-2016-for-mac-64-bit-upgrade

I had Office 2016 running fine on Monterey but I had it running on Mojave and Big Sur migrated over so not sure you could install for the first time also it is no longer supported with updates from MS.