I actually approach with fair a lot. I tend to short hop fair and either fade back or keep going past them to cross them up. You can follow fair with side B or up B before you land. I'll usually do up B if I hit the fair since the startup is faster and you get intangibility, or crossup reverse side B if they shield the fair and then drop shield. A lot of people don't know she has moves she can do after short hop fair, so you can catch them off guard.
You can also full hop fair, which sort of functions like fading upward instead of backward. It's safer when there are platforms though, Lucina has a hard time landing so maybe don't do this if there's no platform to break your fall.
Another option, other than the ones people have mentioned, is to run up and immediately do Shield Breaker. If you run, then let go of the joystick and press B at the same time, you can do it with no lag straight out of a dash (it does require you to be in the full run animation though, so you can't do it during the dash startup). You can even B reverse this by pressing the other direction immediately afterward.
It's honestly not a great option since Shield Breaker has such bad endlag, but it's worth keeping in mind since it is her longest-ranged move. And of course if their shield is low enough, you can occasionally still get the shield break, even after the horrible pointless nerf to its shield damage (why, Sakurai).
But yeah, at the start of a match dthrow > uair/bair is pretty much your bread and butter, especially against fast-fallers who you can't combo with fair or Dancing Blade at 0%.