Let's say there is a space between you and MK that's about as far as the second aerial of your short hop will take you, or about one fair short hop for MK, given that you short hop towards your opponent. Now standard habit of any Luigi would probably be to short hop some combination of aerials to approach, but that doesn't work here. So Metaknight short hops toward you, you short hop backwards and maintain spacing, and Metaknight's Fair hits air. Now Mk is on the ground, and you're in the air still ready to do any aerial, DI forward and close the gap and hit MK. At least that's the basic idea of it. Now what is important is to keep a pattern so only MK is the one being hit, because what do you do if things are off?
First, I don't ever use any move that is going to have any real amount of lag to it. That means, you really only should be using your all of your aerials, your jab, up-angled fsmashes, uptilt, and the slowest move you probably use is the downsmash, which still has quick startup time. I never use luigi's normal Fsmash or his ftilt, downtilt, green missile, or charge any smash moves. All the rest, which Upsmash, and up-B, need MK in a position that is clearly punishable, like that he's falling in general stun after his B moves, or during the lag after Fsmash or uptilt. Also, the landing lag of your aerials are pretty okay, but don't land any fairs that won't autocancel(usually Fairs are fast enough that this won't happen unless you try to fast fall it or do one in the last 1/3rd of Luigi's short hop). The tornado is an exception in all of this- it's good for chasing and punishing lag too, but it's also risky if it gets shielded completely because you do have lag on the end of it. I believe it starts the fastest if you use it just before you land on the ground, and there are definitely occasions I'll use it, but I try to do it when I'm moderately close and will get at least 4/5 of the hits on them, and even more so if their shield is somewhat weakened already because it greatly increases my chance to get a shield-poke with the move. Your tornado can also be completely eaten by Meta-knights Mach Tornado, so that's another reason I'm just at tad more hesistant to use it in this match-up than most.
This may seem pretty basic, but there is no reason for Luigi to give up punishable lag, so don't do it. Second is maintaining the right amount of space. Fireballs are an optimal choice for this. They don't carry a whole lot of hitstun, but they are disruptive to both momentum and attacks. Don't use them too close, because it won't stun long enough to avoid getting punished for the lag. However, when MK likes to just rush you on the ground for a dash attack or grab from too far away, you can just pretty much just stand and toss them. The best place or context for luigi's fireballs, however, are in one half of your retreating shorthop- in the first half, it can catch MK doing a shorthop, or keep him from jumping, and then you can do a falling nair or tornado. Or you can find that you have shorthopped into an aerial, but aren't in a good timing for it for whatever reason, use a fireball. A second effective component for spacing is luigi's Jab. I use jabs a lot to as a reliable choice when I've gotten too close, and usually just after I've landed an aerial, because they come out fast and aren't punishable if the first one hits, and if it doesn't, you aren't likely to have exposed yourself.
Now back to the original idea, to move backwards with MK moving forward. With a double aerial in a short hop, you really start to have the flexibility to handle the speed and range of MK. A shorthop to a dair is usually pretty effective- it either misses, and of course your aerial is fast enough it doesn't matter, or Metaknight just ran into it trying to use his dash attack or grab. If he's run towards you, and for some reason has rolled towards you, bair or nair will hit him. In terms of the best option after you've gotten out of the way of a shorthop approach of MKs, your fair's reach is actually a bit farther than you might have assumed(he snaps his hand out and down, which is ideal for targets in front of and slightly below you, the hand also gets pretty big, it just happens fast), or you can, again, use a nair.
Now then. If MK pulls out his Mach Tornado or Drill Rush, shield it, or if you get above his Mach Tornado, Dair it. Countering these two moves depends simply on how often you expose yourself by doing laggy attacks, and how quickly you react.
You should note how to avoid getting juggled and how to juggle MK. Despite his aerial power, MK can still at least be annoyed with uairs(it is > his dairs). The trick is is to stick to uairs ONLY and to stay almost perfectly below MK at all times. This also helps avoid glide attacks(he can't go perfectly vertical in a glide so you should be pretty much able to avoid a glide attack completely once you are perfectly under him), shuttle loops, the whole deal. I see this "stay directly below them" thing done all the time by CPUs but strangely neglected by players- you can simply upsmash from a dash on the ground even, and it forces the opponent to head for the ledge. As for avoiding being juggled, a combination of a well timed airdodge, fast falling, and falling tornadoes should help.
Now just two remaining very specific insights to handling the MK match-up. One is getting in for the up-angled fsmash. This is your ideal KO move because it is fast, and has a lot of vertical KO power. It's so fast, in fact, that if you were to spam it, it still is likely to hit. Tossing it out at MK in a way that makes him very wary of it, and forces MK into an instinctive reaction to roll behind luigi which can be punished by simply using your up-angled fsmash right behind you.(Just hold up on the control stick and little bit to the direction you want to use Fsmash, and hit over-C to smash, to change directions just tilt the control stick a little bit the other direction) If you're on a little bit of a slope and below MK, the up-angled Fsmash becomes even easier to land and will catch MK in a lot of different attempts to approach you. However, the way I most commonly land the up-angled fsmash in the midst of things is to mix up the usual spacing and simply walk just a step towards MK and use an up-angled Fsmash while luigi slides a bit. It should happen fast enough that it catches MK by surprise while he moves thinking you'll do something else.
The second thing is recovery. MK gets his best kills(possibly below 100%) by using a reverse shuttle loop either while you are above the stage(easy for him to do) or while you are coming in from below the stage(harder, he has to grab the ledge). So to solve this problem, learn to sweetspot the ledge from a charged green missile(aim a little lower than the ledge itself). A chance of misfire/strength of a fully charged over-B is pretty annoying for MK to get hit by, too. Between this and not exposing yourself in the air by never doing anything more than a shorthop and at worst use a fast fall airdodge back to the ground(ie. no double jump from the ground, no full jump from the ground, no botched attempts to juggle), MK's chances to KO you DO get reduced considerably. Then it becomes all about strong hits from his Nair and dsmash. This adds to how long you will live per stock a lot, and really really helps.