If a centaur were to have one set of organs, how would they be placed and what sizes would they be?
What limits would there be on the centaur's body?
A horse's digestive tract is 3-4 times the length of a human digestive tract because horses are able to survive by eating grass, which is very difficult to digest. Centaurs as we typically picture them can’t possibly live on grass, because human hands and mouths are poorly adapted to pick and eat grass. So they pretty much eat human food. You could plausibly keep elements of both the human and horse digestive tract by having food first pass through a human stomach (which breaks down food using hydrochloric acid) and then a horse stomach (which uses fermentation) before going to the small intestine. Arguably, this doesn't even violate your “one set of organs” rule, since even though these organs happen to share the same name, they work in completely separate ways. If you want to use only have the human stomach and restrict the centaur to eating only food that humans can eat, you are looking at a smaller, slimmer horse because you don’t need the extra space.
Horse lung capacity is around 50 liters. Human lung capacity is around 6 liters. Clearly, if we want the centaur to run like a horse, we need to go with the horse lungs. By we need a windpipe big enough to serve 50-liter lungs. Since the human lung capacity is more or less redundant, let’s assume a significant portion of the centaur’s human torso is occupied by a horse trachea which is 5-6 inches in diameter. However, the shape of the human torso is really defined by lungs. Without lungs, there is no reason for a rib cage, and without a rib cage the centaur just a skinny little spine from hip to shoulder. I recommend that you let the centaur keep a set of small (say 3-4 liter) human lungs in addition to (or connected to) its horse lungs. In addition to making the centaur look right, these could be provide the fine breath control required for understandable human speech. It’s a plausible guess that powerful horse lungs wouldn't be capable of speech.
Like with lung capacity, we are going to need at least a horse-sized heart to power the centaur. Rampant speculation leads me to believe that it will actually need to be maybe twice the size of a typical horse heart in order to supply sufficient blood to the centaur’s brain, which is about 1 meter feet directly above the centaur’s heart.
In a relaxed state, a horse's projects out from its shoulders a fairly shallow angle less than 135-degrees. A centaur's human torso is typically imagined as projecting up from its withers (horse shoulders) at a 90-degree angle. This means that you need some really beefy muscles around the withers to keep the torso from pitching forward and to stabilize it side-to-side. These extra muscles will probably form a large knot around the shoulders and then slope down toward the horse’s rear, giving the horse-half’s back a much more slanted profile, something like a cross between a giraffe’s back than a horse’s back. It might be possible to ride a centaur, especially since you can hold on around their human torso, but it wouldn't be easy or comfortable without a specially-designed saddle.