Hello. My name is Marc. I am a licensed attorney and I will be happy to answer your question. I can imagine your anxiety over this. If you have no convictions, you should be ok.
I know that doesn't sound 100% reassuring. The main question is whether you were arrested in relation to the felony. The fact is, even if you have no convictions, arrests typically appear in any criminal background check. It will show the date you were arrested, jurisdiction, and charge. BUT, it will also show the disposition (e.g., dismissed).
If you were not arrested, the existence of the case may appear, but it is less likely.
In your case, if you were arrested, the record will likely show the matter as pending. It's frustrating, I know. It's a black mark on your record even though you haven't been convicted. However, that's why most job applications ask job seekers to list only convictions. The bar/restaurant where you are applying for a job may well not care that you were arrested, as long as you weren't convicted. As for any enhanced security requirements for working in an airport, federal regulations would only bar you if you were convicted. Even then, you would have had to be convicted of a "special offense", which includes pretty serious crimes.
In sum - your prospective employer might find out that you were arrested. If so, your employer might not care. And the fact that the job is in an airport does not make it any more likely that you would be denied the job.
Overall, I would say the odds are in your favor, so I hope this eases your anxiety somewhat (I wish I could guarantee a favorable outcome, but I'm afraid I cannot).
I hope you found this info useful. If so, please be sure to rate my answer, since that is the only way I can receive credit.