You must beat the entire game in one sitting on the Original difficulty setting in under 50 minutes to obtain the Superhero! achievement. My suggestion is for you to get familiar with the game and it's levels first by playing on easy or normal which will not only show you how to get through the levels, it will also show you tips on how to beat the bosses.
After you are knowledgeable of the levels and how to easily defeat the bosses, try playing on original difficulty to get a feel for the difficulty changes. Then view these videos and practice individual levels to memorize the shortcuts. When you feel comfortable, give it a shot by starting a new game on original difficulty.
Below I posted speed run videos of the nearly identical Genesis version. I personally used these videos to learn the various shortcuts which helped me obtain a 33 minute completion time. The easiest levels in my opinion to shave valuable time are New Junk City, What The Heck, Level 5, and For Pete's Sake.
A couple of additions I would make to the videos are that you can no longer go straight from Billy The Bin to Chuck in New Junk City as you now have to play through New Junk City II. There is a time when you need to choose one of two routes labeled "Stud" and "Wimp". Taking the "Stud" route is quicker. I can't get the tactic in Snot A Problem to work but would love confirmation from somebody if it works for you. You'll also see the guy in the video shoot the end boss for a couple of minutes. If you whip the last boss as opposed to shooting it the battle will only last you a second. ;)
I didn't create these videos nor do I take credit for them. I'm just posting them to help you all out!
Good luck everybody!