Some tips for the revamped boss fights:
Sheep Man - In Hard Mode, his lightning strikes will go off almost instantaneously, one right after the other. Also, the first strike will send sparks across the floor as well as the last one. To dodge this, just move wherever there is the most space between clouds and jump over the sparks. He will also send sparks across the floor whenever he hops backwards, so watch out for that. Sometimes he will just stand in place and shoot out a spark at you in a wave pattern. To dodge this, just jump over the low ones and let the high ones fly over you. After a low one, the next one will be high, and then low, and so on.
Pump Man - Only one real change to note here, which is that after releasing his Water Shield, he will throw the handle on his head at you like a boomerang. This new attack HURTS, so concentrate mostly on dodging that attack. Other than that, the same strategy from Normal mode should apply.
Solar Man - This guy can be a pain. The flare on his head starts out at full power on Hard mode. My strategy for this fight is to look at the arena in 5 parts, with each elevation change marking a different part. For each part that Solar Man jumps over, he will shoot down a Solar Blaze. Keep in mind, he could potentially jump all the way from one side of the arena to the other. Throw up a Water Shield, study his trajectory as he jumps, get near his landing zone, and hit him with the shield as he lands. If he does jump all the way to the other side, you could land an extra blow by putting up another Water Shield and jumping barely high enough to hit him with it. If you try to run your shield into him after he lands on one of the lower parts of the arena, he might block it with the flare on his head. On Hard, he also has a new attack where he stands on one side and shoots up 3 Solar Blazes in succession. Considering he just stands there in the most vulnerable part of the arena while he does this, you can use that to your advantage and just bombard him with Water Shield bubbles.
Chill Man - He moves a little faster this time, he shoots out 3 Chill Spikes with his signature attack instead of 2 (one low, one high, one in the middle), and he also has a new attack where he jumps into the center of the arena and shoots out 5 Chill Spikes in a semicircle. When he does his new attack, stand in the far corner (or maybe a step or two away) so the spikes don't hit you. The spikes can be destroyed with the Solar Blaze, so shoot one out after that attack.
Nitro Man - Oh dear God, this guy gave me nightmares. His engine and handling have been upgraded this time, meaning his motorcycle form moves faster and also turns around faster when you jump over him. To avoid him this time, you'll want to do lighter hops over him, so you'll be able to jump again when he turns around. Another thing to note is that Nitro Man is the only boss to be frozen by the Chill Spike, so use that to your advantage by freezing him until he changes forms (to avoid having to worry about excessive Wheel Cutters on the screen) and then laying down one or two spikes in preparation for his charge.
Commando Man - One new attack to note here. Sometimes he will jump up and pound the ground, sending shockwaves all the way across the floor Use the Wheel Cutter to scale the wall to dodge it. He is still not that hard, so just rush him with Wheel Cutters.
Blade Man - His Triple Blade has turned into a Quintuple Blade this time (he shoots out 5 at a time instead of 3), and he has a new attack where he headbutts you when he touches the ground. To avoid his Quintuple Blades when he shoots from the walls, stand in the far corner and start moving to the other side just after he fires. If you successfully dodge it, his shots from the ceiling shouldn't even touch you as long as you keep moving to the other side.
Strike Man - He has one new attack where he jumps into the middle of the arena, spins around, and shoots out 4 Rebound Strikers at once. This can be tough to dodge, but he is so weak to the Triple Blade that he should still die before he can even do it. The boss is easy, but his stage is probably the hardest of the 8 Robot Master stages on Hard Mode. The Water Shield should really help in getting past the locker room enemies.
Wily's Weapons Archive - Same as in Normal mode, except the 9 bosses attack just a little bit faster. Same strategy for Normal applies here.
Crab Puncher - He moves in a little closer this time leaving you less space to move around. Also, when he does his bubble attack, he shoots out 3 rounds instead of 2. Other than that, same strategy for Normal applies here.
Block Devil - All I can say is that you had better have the block pattern memorized, because the blocks now move at lightning speeds.
Wily Machine - Nearly identical to Normal mode, except the first form's homing missiles tend to stay on the screen longer if not destroyed.
Wily Capsule - This thing's attacks will move faster. The side-to-side attack tends to go back and forth longer, so be sure you know where the safe spots are. The homing attacks will now charge at you 3 times instead of 2 and also move faster.
Once you know what to expect for the new versions of the boss fights, D0ntTaseM3Bro's general tips should be enough to give you that extra assistance to conquer what is, in my opinion, the hardest Mega Man game since MM1 (on Hard mode, that is).