Cappadocia is a landscape that happens to have an aviation industry attached. Around a hundred balloons fly on a good morning, all of them between roughly 05:30 and 07:00, and none of them if the wind is above about 12 knots.
That single constraint should shape your entire trip.
Book the flight for your first morning
Roughly one morning in five is cancelled. If you schedule the flight for your last day, a cancellation means it does not happen. Schedule it for your first and you have two or three chances in hand.
Operators refund cancelled flights, so there is no financial reason to do it the other way round — only the temptation to sleep after a late arrival.
Fly on day one. Keep the rest of your mornings as insurance.
The best months
April to June and September to October have the highest flight success rates, the mildest walking weather and the clearest light. July and August are hot enough that midday walking is unpleasant, though the flights themselves are reliable.
Winter is genuinely beautiful — snow on the fairy chimneys — but cancellations rise sharply and several valley paths become slippery mud.
What to do with the rest of the day
You will be back at your hotel by eight, having been awake since half past four. Sleep. Then walk in the afternoon, when Rose and Red valleys turn the colour they are named for.
The underground cities are best in the middle of the day, when the surface is hot and eight storeys underground is not. Derinkuyu is deeper and narrower; Kaymaklı is roomier if enclosed spaces bother you.