Simple approach might be:

Create a starter program that checks for the existance of a semaphore file in a specific location. If the file there it runs your program and then deletes the semaphore file.

Set this starter program up as a scheduled job under windows to run at some acceptable interval (like every hour or whatever is frequently enough without being too frequent)

On your local machine (the one with the mapped drive) your program can create the semaphore file on the remote machine when it needs to make the remote application run.

The semaphore file can be as simple as a single space because its existance is the message and not its contents.