It looks like a fairly standard size lithium battery in the clasp. I would guess it would last for a few hours powering the radio.
I had a Sinclair TV many years ago, that ran off a unique credit card sized lithium battery that cost about 10% of the cost of the TV and lasted for only 6 hours! Fortunately it also had a jack plug for an AC adapter. But like some disastrous Sinclair products of the time, it was something that impressed with what it was possible to do, but ultimately wasn't that useful in real life.
Still, that radio watch looks appealing in a showing off to friends way.