"One second per 300 billion years" is really a statement about the bandwidth of the signal that the clock generates, translated for lay readers. We don't know what the actual bandwidth of the signal is, since it's not stated in the article. Bandwidth is measured in Hz.

In order to measure the bandwidth of the signal (and how accurate the clock is), you have to measure the clock over a period of time. The shorter that period of time, the worse your measurement will be. The smaller the bandwidth, the more time you need to measure it.

Note that a signal with 5 Hz bandwidth and 5 Hz center frequency would make a very terrible clock, while a signal with 5 Hz bandwidth and 5 GHz center frequency would make a very nice clock. The bandwidth is the same in both cases, and the amount of time required to measure bandwidth is the same, but they make very different clocks.