Lactic acid isn't even what causes the burn. Technically it's lactate. In any case yes, lactate dissipates completely from the muscles in about an hour, maybe less for the well-trained. The lactate burn happens because your muscles aren't yet good at either using it for fuel or removing it from the muscle. The traditional method of lactate conditioning is to get in a long hill and go over your lactate threshold, get a good burn going - you'll be panting - then back it off to your upper aerobic zone - you'll be breathing deeply - then take it back up again and repeat until you can't anymore. Not the most fun anyone had, but it works. It's just a matter of conditioning.
Even more technically, it's now believed that it's not actually lactate that's making the muscles burn, but whatever it is, it's created at the same time as lactate, so might as well blame it on lactate because it's in the language now.
Muscle pain the next day or more usually, on the second day, is called DOMS, delayed onset muscle soreness. It's from microtears within the muscle. They take time to repair. In general one can hit it hard maybe 2 days in a row, then one needs a day or two of recovery or easier rides to allow the microtears to heal. If you keep hitting them before they heal, you get weaker, not stronger.
OTOH, sore legs aren't necessarily from DOMS, at least in my experience. Sometimes they're just sore. If I get on the bike and pedal and they still hurt after 1/2 hour, I give it up and pedal easy back home or just get off the rollers. In general, if sore legs on the bike are a consistent problem, you're not getting enough protein to heal them up. More usually they loosen up and feel OK after 1/2 hour of fighting the urge to give up and go home.
I ran a crew doing very hard but intermittent skilled manual labor. I found that going to the gym and working out really hard a couple times a week made a huge difference. Focus on the muscles which get sore. Beat the crap out of them.
As I say, supplements are a feather. Training is the hammer. If you're weak, train. That said, try electrolytes before bed and pickle juice for cramps. Not too much magnesium! My wife likes potassium. But that's for cramps, not sore legs.
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