Does apple remote desktop on macOS 12.0 Monterey works with Big Sur

The current version of ARD is 3.9.5 and requires macOS 11 Big Sur or later so yes, it will work with Big Sur. It can be bought from the Mac App store.

The App store will advise you with each software about compatibility and version numbers.

You're very welcome, and welcome also to click on the solved or helpful button if you think it is appropriate.