I think it is the same join timer as the Slice is Right. Generally speaking, if you are hunting for that achievement, you should duck into the Gold Saucer at the times when it can pop to see if it shows up. In my experience, if you stand in the "most safe spot," you have a 40% chance of success. It will take about 25 runs with average luck to get WTFungah III completed (if that's your goal). Back when I did it, this GATE only popped at X:20. So just check the Gold Saucer every hour on the 20s. I think they increased the spawn frequency in a patch at some point where it might spawn at other times, but I don't know that information off-hand.

In regards to the safe-spot, check this reddit thread: https://www.reddit.com/r/ffxiv/comme..._gate_aka_afk/

As of 2022, it was still accurate.