Use rsync to automatically copy a file to another folder
You could use, Automator and Folder actions to watch the folder and run the script.

When launching Automator and choosing New, one of the options given is to make a 'Folder Action'. At the top of the screen after selecting Folder Action you are given the option to choose a folder to 'watch'.



What I think you would want to do in your case is to then select 'Utilities' on the leftmost column, and then drag 'Run Shell Script' from the second column into the workflow window. 



Here is where you would write whatever script you would like to run when changes are made to the folder. In this case your rsync script.