I'm sorry to say, No.
Any timer, either on an ASP.NET page or in an Ajax method, are tied to be run while in an opened browser.
The only way I can think of would be if you could have a script available, that takes care of sending those newsletters, and also be able to manipulate the servers scheduling agent.
What I'm thinking here is that when you click the button, you create a new schedule that executes said script.
But unless it's your own server, I doubt that any web host would allow you access to Windows core services.
If that was possible, the scheduler would take care of sending those newsletters for you.