The Sixth Floor Musuem, which owns the remnants of the Hertz sign, doesn't know how it was powered but assumes it was wired into the building's electrical supply: https://www.jfk.org/collections-archive/hertz-sign/.
At the time, the TSBD Hertz sign was one of only three such signs in existence.
Signs of that sort during that era typically did not have backup power.
Obviously, the clock would have to reset or be reset somehow after a power outage.
Whatever you think the significance of this issue is - I see none, but whatever - you will never have a definitive answer.
Thanks, Lance, for reminding me that the Sixth Floor Museum website includes
THIS PAGE all about the Hertz sign atop the Book Depository Building.
For those interested, here are a couple of pictures showing the other two large billboard signs that occupied space on the roof of the TSBD prior to 1959. Click to enlarge....
The
"U.S. Royal Tires" sign (this photo is from 1948):

The
"Ford: Built In Texas" sign, which went up in June 1953:

Click the logo below for many more Dealey Plaza/TSBD images, dating back to the 1930s:
