There are no special requirements to build an airport, they just cost a fair bit so unless you're specifically just going for this achievement it's a good idea to make sure you have a decent economy up and running before you go out of your way to buy one, if you build a "Diplomatic Ministry" and enact the "U.S. Trade" edict then the cost of building an airport will be halved.
They can take awhile to build so be sure to send your avatar/dictator over to the construction site by holding "RT" and then pressing "B" to select him, then pressing "A" on the airport construction site, doing this will speed up production and make the builders work faster!
As soon as the airport is built the achievement will unlock, you don't even need to have it staffed!