I reassembled the alarm.
Before doing so I soldered in some wires to the positive and negative battery leads and routed the wires out the back to hook up to the solar panel.
Tied the wires in a knot to stop them from being ripped out and/or putting stress on the soldered joints.
Drilled a hole in the back of the case for the wires to exit the case.
Then reassembled the clock.
Next up I'll have to find me some sunshine to test out weather the solar panel is actually going to charge up these batteries or not. Since the clock uses so little power it shouldn't take much to keep a sufficient charge.