Doing FATES solo might be a bit on the slow side, but the point is you can still earn xp while waiting on your dungeon queue. Just keep an eye out for the double xp ones that pop up from time to time, and again for the special enemy that can show up and give you a double xp buff for your next FATE.

It's supplemental xp and is faster levelling than queueing for a dungeon and tabbing out until it pops. As long as your chocobo can cast choco-cure, you can solo any of the non group specific ones with ease.