Use the Armspin style and nothing else.
You will need all the upgrade modules you can get from the side rooms and its always nice when they spawn instead of a shop on Hard difficulty. Plus, with the Armspin style, none of the side rooms are going to contain extra gems.
Shop prices are super expensive on Hard, meaning you will probably be buying toward the end levels.
The key to success is being agile and cashing in the 25 or higher combo, as this is going to give you 100 gems +1 ammo charge and +1 health.
The essential upgrades for Hard is Gem Powered, Gem Attractor, Laser Sight, Drone, Reverse Engineering, Candle and Apple. (And Knife and Fork if you get it early to mid game)
Other useful upgrades are Popping Gems, Gem Sick and Youth.
The first area is the toughest in my opinion. Getting through this one without a scratch is going to require skill and luck.
If you move too slow, the red eye enemies will come from above and if you move too fast, you might land on a red enemy, whose numbers are going to be considerably higher than on the normal difficulty. The only other addition to this area is the new left to right flying mini shell/clam guy, who is super easy to stomp or avoid.
The second area is not too hard.
There will be a larger quantity of enemies, but they are slow and easy to kill.
The thing that'll require getting used to, is the timed spikes. Unlike normal mode, the red blocks spike up every few seconds, so you have to time your descents more carefully.
Don't forget to stomp the torches as they are essential for controlling your descent and replenishing ammo.
The third area is super easy.
There will be more squids, which is useful if you have the Knife and Fork upgrade and instead of clams or barrels, oxygen bubbles will be on their own, all over the map and not in a batch, so that means more oxygen collecting.
The fourth area is super easy if you have the Gem Powered and Gem Attractor upgrades.
The new addition to this area is that it will contain all of the flying and floating enemies from all of the other worlds. So you will have ghosts, skulls, bats, turtles, eyes, squids, piranhas and so on.
With the mentioned upgrades, you can literally spam fire, because the gems and floating objects will replenish your ammo.
Ant lastly, the Boss, who is barely tougher than on normal mode.
Everything is mostly the same. Make sure you have a laser during the boss fight and all of the mentioned upgrades.
There will be more adds and the boss will shoot fireballs two times in a row, so remember that and keep your distance.
That's all there is to it. Just practice, be patient and ye shall succeed.