While the ratio and low number of completions may make this seem like a daunting task, the game is really not as hard as it first appears. By doing the levels in the right order, using the right gun at the right time and utilizing various cheesy methods, this is indeed quite a simple task. Not easy, certainly not something anyone could do, but nowhere near as hard as the TA score makes it out to be.
The only gameplay difference between Normal and Old-School mode is that health pickups now only restore one health point instead of two. Not a huge deal, so feel free to practice the final level as much as you want in Normal before attempting this. I highly encourage going for both endings in Normal before trying Old-School, just so you have a decent grip on how to make it through Temple of Guns and its boss as safely as possible.
As for level order, you should go:
New Gun City > Mount Scorcho > Rhulke > Fett's Train > Forest of Chad
Feel free to switch the first two around if you think Scorcho is harder. I personally die more often on NGC, so I started with that. Make sure you get every Torpeagle seed bag (one in each level, all available with no extra guns). This is extremely important, as you need at least four shots with it to make the second phase of the final boss doable in a safe-ish way. More Dual Magnum and Back-a-Pack-a-Pult seeds also do not hurt, if you can get them safely.
After the initial two hard levels, the remaining three should be fairly easy. The Dual Magnums from Rhulke completely destroy Fett, and picking up more seeds for them in his stage means that you can also make short work of the Spider Queen, who can otherwise be a bit of a scary fight.
EDIT: Some new boss weaknesses have come to my attention recently:
- Rhulke is a joke if you use the Drill Gun since it can hurt him whenever, not just when he's in the air or stunned after a charge. Two seeds are enough to kill him.
- Dennis (NGC boss) is extremely weak against the B-Uzi, two clips are more than enough to kill. Thanks to Chizzy for that discovery.
- The Spider Queen takes one whole health chunk of damage from every Drill Gun shot. If you jump and shoot up, it's pretty easy to hit her until she starts moving fast, at which point you may want to switch weapons.
Use those tips as you see fit.
For Temple of Guns, don't hesitate to use the Torpeagle and Backapult to destroy particularly pesky blue/red spike enemies. Your seeds are refilled when you enter the temple and before the final boss, so you don't need to worry about running out.
Here is a video of my full Old-School playthrough with no deaths. Study it, learn how to make hard obstacles easier, and this should be very attainable with a bit of practice. Note that this run is by no means as safe as it can be, so feel free to play more carefully if you prefer.