From your description, I'm guessing that you hear a startup beep and see the display blinking with one of the startup messages--probably "pwr rset" where the "w" looks like a "u" with a bar over it. Is that correct? If so, my first impulse would be to test the buttons (discussed later).
With respect to the schematic, there are Eagle files for the main board in the xmas-icetube GitHub repository
here. I'll try to attach a PDF, in case that's easier.
The buttons and alarm switch circuit is pretty simple: the buttons/switch just connect a pin on the microcontroller to ground. There is a pull-up resistor for each of the microcontroller pins within the microcontroller. Those resistors are enabled for the buttons/alarm pins when the clock starts, and the pull-ups are disabled when the clock is off or sleeping on battery power. When the buttons/switch conduct and the clock is powered, the logic level on the corresponding microcontroller pins should be pulled down to ground. When the buttons and switch are open, the corresponding microcontroller pins should be pulled to +5v.
So to test the buttons, I would verify that they do not conduct when not pressed, and that they do conduct when pressed. Then check to make sure that the voltage at the corresponding microcontoller pins is dropping to ground when the buttons are pressed and rising to +5v when not pressed.