Is it possible to use AutoIt mouse movements in a remote, cloud-based Windows server?
It depends on what you really mean.
If you mean you want a compiled script to run on a remote server even when you're not logged in, you can do that. You just need to write the script and test it from the server, and then compile it to a .exe file, and then set up a Windows scheduled task on the server to run that .exe file whenever you need it to run.
If you mean that you want to run a script from your local machine that does stuff on the server, you can't do that, unless you tell your script to include the steps of logging into the server each time. AutoIt is a GUI automation tool; it can't remotely access another computer on its own, unless you tell it to do so via the UI you have access to on your local machine (or rather, the machine where the .exe script is run).