Price: Don't remember. I fired off a message to the guys in manufacturing, I'll let you know if they have a price. We go through quite a few of these annually, so we have a bulk discount when buying them.

How: Magic, I guess. Also, I found a link to the ones we use: https://www.microchip.com/en-us/products/clock-and-timing/components/atomic-clocks/embedded-atomic-oscillators/csac

When we prepare for deployment, they're connected to a docking unit which provides a clock pulse derived from a GPS signal for high precision. Then the battery pack is attached, the subsea unit is assembled, and everything is deployed, usually via ROV. It's important that it's kept powered the entire time upon retrieval docking, as we can then calculate a linear drift value and correct for this in the recorded data.

UPDATE: We bought "Maaaaaaany thousands" of them in 2018, and we paid 1850 USD per chip. Come to think of it, I remember hearing claims back then that we'd bought 75% of the worlds supply.