In order to finish Arcade Mode on Hard, you'll need to start on Hard, and then use 3 "special techniques" on the bosses of world 1, 2, and 3. (Otherwise it will automatically drop you into Normal difficulty after the boss.) You'll hear a distinctive sound effect after doing the "special technique".
World 1 boss (giant lumberjack machine): sometimes he'll toss sets of dynamite at you. You need to knock it back into his face. I recommend standing on the second platform and whirling with B.
World 2 boss (wolf king): Pretty easy, you just need to knock the floor out from under him. Go up to the second floor and wait until he joins you. Then drop down on the side and hopefully he'll run over the floor block; destroy it as he's running over it.
World 3 boss (Axel Gear): This boss is tough on Hard, and the "special technique" is hard too. Good luck. :) When he fires a little purple homing orb at you, you need to circle around so it ends up hitting him. Tricky! Thankfully, lives aren't a problem as long as you restart whenever you die.
It's not hard at all to finish "in a single playthrough" and get this achievement, because you're still allowed to restart levels if you lose too many lives. (Aways, always restart if you die to a world boss!) Just don't ever exit to the menu. Expect the playthrough to take 3 hours or so, depending on how good you are.