Falco is probably my best Sheik matchup, so I'll offer some tips.
First, the other guys told you to stay on the ledge against Falco. I think that's a mistake. You messed up your second attempt because you just fell off into an f-air, but if you'd fallen into a double-jumped f-air, you would've hit him, and there would've been no way for him to recover. Be careful about it, but every chance you get to pressure Falco off the ledge and get the early gimp kill, don't hesitate. Falco's got the advantage over you when he's on the stage, so you'd better take the advantage off the stage. An experienced Falco won't try to recover in the trajectory of Sheik's needles, anyway. And when he catches on to the f-air off the ledge, he'll start trying to go straight up from below the ledge. That's when you wait a bit, then fall on him with an n-air, so you take the hit from the Firebird and get popped back up to the stage, but he dies. Then he'll start trying to hug the stage and recover sooner, at which point you should fall off the ledge with a b-air. Eventually he'll learn to tech it, at which point you should generally stop doing it, because he's usually beat you back to the ledge for an easy edgehog. But in tournaments, all these things are useful, because no one has time to adjust to your style completely.
Also, Drephen mentions staying on the ground. Definitely. A good Falco isn't going to get hit by your aerial needles during his approach; he'll space it such that, at best, you'll hit his laser. How do you stay on the ground against Falco? Pretty simple, actually. First, learn to powershield lasers. It's not hard, it's not unreliable, and even one powershield, as long as it looks intentional, can be enough to break a Falco's confidence in his approach and create openings when he hesitates. Just learn to feel when your powershield is going to work, so you can be ready to chase it. Most Falcos stop dead in their tracks for a while after they get hit with their own lasers.
Another thing, which is always important with Sheik, is spacing. Don't be afraid of the lasers; they aren't what kill you. It's keeping yourself immobile to try to not get hit by them which will get you killed. So instead, keep the match at mid range. Your attacks are faster and generally have a longer range. Jab > f-tilt, even on shields. Then get in closer, jab > up-tilt. The second hit often catches people off-guard. A very important idea is to wait for the Falco to get nervous and go for the laser at the mid range. That's where you just f-tilt him out of his short-hop, before the laser.
As for getting around the lasers, more tips:
Learn to short-hop over them and waveland in the direction they're traveling, and they'll never hit you. Duck under them, because most Falcos don't get them low enough to hit you, then wavedash backward between them (while still crouching) to get away from the approach. If there are platforms, wait for the approach, then run away, jump toward the platform, throw some reverse needles, and land on the platform. This is completely safe.
In general for the matchup, just make sure that, when Falco leaves the stage, he doesn't get back on. Otherwise, play it safe and generally more defensive than you would with other matches.
(I felt like giving an speech.)
kekekek
