Curious to know if there is a specific reason they fire it at 1pm. Noon would make more sense to me!
Edinburgh Castle's tourist attraction, the One O'Clock Gun, originated as an audible version of the Nelson Monument time ball. The Nelson time ball has dropped at 1:00 pm ever since it was installed in 1853. Accordingly, the One O'Clock Gun also fired at 1:00 pm. Anything else would have been potentially confusing.
Time balls were important for maritime towns as a means for ships to set their clocks to the correct time. This was important for maritime navigational purposes.
In England, time balls typically dropped at 1:00pm. This seems to have begun with Greenwich, which dropped its time ball at 1:00pm from 1833 on, because its astronomers would've been busy with observations at noon.
One o'clock was chosen as the perfect moment, in order to give the astronomer time to finish his noon observations before the ball was dropped.
- Dibdin, Tim. Time out guide: Edinburgh, Glasgow, Lothian and Fife. Penguin, 2000
It would have made sense for all the port cities to pick the same time for updating clocks, so as to prevent confusion.