How do you get into the cyclone on Final Fantasy II?

You can only enter it when the story tells you to go there. If you have not met Ricard yet or his wyvern, you aren't up to that point yet.

The Cyclone will first be mentioned in the story after Minwu unlocks the Ultima Tome for you in Mysidia Tower. The mages of Mysidia will be telling you to hurry to Fynn (even though you can just walk the whole way). To access the Cyclone, use the Pendant on the big, blue mirror in Fynn Castle to summon the last wyvern, which will fly your team into the Cyclone.