Just use three small diodes in series to produce the 1.5 volt reference voltage with a 10K - 15K resistor for the voltage dropping component. After that use a large value low voltage capacitor as a backup power source for line power dips and outages.
5 components and thats it all salvageable from almost any common circuit board.
This is what I suggest (others have suggested a similar arrangement).
I suggest using two diodes, I notice that TCM has suggested using three.
However I think that three may be too much.
I just did a quick test using a bunch of 1N4007 diodes and got a forward volt drop of 0.71 to 0.72 volts across each diode.
Using two of those diodes in my proposed circuit would give your clock a supply of 1.42 to 1.44 volts which should be adequate for normal operation.
Better use a 1.5V battery, because the current range in your case is in uA. You wont save huge using an adaptor.
But, if you still want to use a adaptor, then 7802 would also work. 1.5 or 2 wouldn't make a large difference. LM117(/LM317) is a better opt, but if you are unable to rig up the circuit/if you want it much more fast, then you can go with 7802IC. But take expert opinion, i might be wrong.