Is your intention to not allow any of the buttons to turn off the alarm sound until you enable it?
Without knowledge of how the circuit in the clock works, the only way is to switch off one of the wires to each button.
If all the switches have one side connected together, then you can stop them all with a single relay.
If you don't want to use relays, then the best would be opto-coupled FET's (AC type or wired correctly to be polarity insensitive).
The contacts of a relay or opto-fet are isolated and so don't care what the circuit looks like. You would simply wire a relay contact (Normally open) or the output of the opto-fet in series with one of the wires to each switch (or if the switches are on the pcb directly, you would need to cut the track to one side of each switch and wire the relay across the track that you have cut. Then you can control the functioning of the buttons with the Arduino.
If you intend to actually operate the clock functions that normally would be done by the manual buttons, then you would wire the relay contacts or opto-fet output across each of the switches.