Everyone seems to have mega man as one Luigi's worst MUs while this is true, I don't think it's as bad as everyone may think. I'd say -1. Though I'm a bit more expierenced than the mega man I play. I say Luigi's problems are approaching, killing, and offstage. However, Luigi does have a good offstage game himself against mega man.
1.Approaching- Pellets/Lemons are annoying plain and simple. They beat fireball and cyclone, and because of Luigi's slowish ground speed and traction, hard to get through. You can get in by rolling, but try not to roll directly into him, as he will nair and pretty much reset the situation. Try to get as close and you can and jab, or even ftilt, just to hit mega man. Due to pellets he doesn't have a jab of his own to combat it with. Then you can try to go for the grab
2. Killing- Mega man is extremely difficult to kill vertically. His small size and weight, make down throw to cyclone practically useless. However down throw to bair seems to work well on him, as does down throw to dair. He falls quickly, so he's a bit easier to get the sweet spot on. Plus you can get come good combos at low percents. Grab>fair>regrab..etc. Also mega man has poor landing options, try to bait a laggy hard knuckle or an airdodge, and you up smash can up b him (I think a lot Luigis forget to use this because of down throw).
3. Offstage- Mega man can bair Luigi out of most of his recovery moves. Like most mus recovery very high or very low. There is invincibility in the middle frames of Luigi's up b so you won't get hard knuckled. The good news is Luigi can gimp him with cyclone. Due to mega man's fall speed, he falls out cyclone easily, and due to his poor air speed he has recover pretty much below the stage. Fall down and cyclone, and he's pretty much done. Yes rush is a good recovery move, but not as good as Sonic's spring.
Overall, it's a tough and annoying MU. It's a battle of attrition. If mega man can patiently keep
Luigi out and forces Luigi to over commit he wins. But if Luigi can pressure mega man in the neutral he wins. They are polar opposites, one's weakness is the others strength and vice-versa. It's not like brawl falco where you had the laser and illusion camp game, and all the crazy jab stuff to deal with. Below I posted a stream where I played my friend Burst each other in two sets. It was a charity tournament, but 2nd place got a Greninja amiibo so we definitely were trying.
http://www.twitch.tv/havenshine/b/664299844
Winner's Semis: 1:41:00 vs Burst
Loser's Finals: 2:35:00 vs Burst