juno,

symptom ng bad contact sa battery terminals ... or, very unlikely, an internal defect of battery, a crack or anything that could prevent delivery of power intermittently

but the clock's strange behavior is a real puzzle

meron ka bang malaking capacitor sa sound system mo? even a small capacitor might be capable of powering the clock for a minute or so ... to verify this, disconnect the battery, turn on ignition, turn on headlights and/or plug the lighter to quickly discharge whatever is trying to provide power to the clock ... or simply disconnect the battery overnight then check the clock in the morning (after reconnecting battery) ...

or if you know how to use a voltmeter, disconnect the battery, turn on ignition, then check the battery terminals with the voltmeter, if you get any reading other than zero, this indicates presence of a power source and if it is a capacitor, it will eventually discharge (which can be obeserved with the voltmeter as its voltage will slowly drop to zero)

note that the troubleshooting procedure above is only to find out why the clock is behaving strangely, not for the "intermittently dead" problem

good luck!