No, that should be USB controllable, so it would be available from Internet & by schedule. Radio module is not enough.
I think we need more information, and I think you need to break the project into parts.
I understand you want to turn something. That isn't hard. Lots of ways to do it with Arduino in control of how much it turns, when, how far, etc. The best way to set things up will depend on what you want to turn.
That's one part of the project, as I understand it.
The other part is: Once you have an Arduino which can turn the thing... How will it be given commands to do the turning. Again, we need to know the sort of turning you want to do.
You mention internet, so I assume you want to connect to the Arduino, give it the commands to "turn left 80 degrees" or "start turning clockwise at one degree per minute for the next half hour" or whatever from a webserver somewhere out there on the internet.
To do this, you need to do all the things in an ArduServer. See ArduServer.com When you have an ArduServer working, you are 95% of the way to making the thing I've guessed you are trying to make.
One of the things an ArduServer needs is an "always on" internet connection. Do you have this? (You don't need a "static IP address", if you've heard of these and are worried). If you connect to the internet with a "dial up" connection, in other words by the phone, and you can't use the phone to talk to someone when you are using the internet, then you will have trouble creating what I've guessed you want.
Good luck! Sounds like a fun project!