I have a character (who is human) who, for his own reasons, decides to act bat-like and sleep hanging upside-down.
Handwave the feet/attachment issue.
Which parts of a human body would be affected by sleeping upside-down for 8 hours at a time?
Related: How would a humanoid need to change to sleep upside-down?
Not a single part, the whole body will suffer.
Hanging upside down a person was a technique of torture used in Japan, called Tsurushi:
Tsurushi (Japanese: 釣殺し), or "reverse hanging", was a Japanese torture technique used in the 17th century to coerce Christians ("Kirishitan") to recant their faith. [...]The technique was said to be unbearable for those submitted to it. The body was often lowered into a hole,[...] Typically, a cut would be made in the forehead in order to let blood pressure decrease in the area around the head.
Our circulatory system is made so that the legs can withstand the higher pressure resulting from being below the hearth, while the head is not.
it was found that, unlike the veins in the legs, the veins of the upper body do not have the valves that prevent the reflux of blood.
Thus not cutting the forehead was even worse: (graphical description)
When torture was further prolonged, the victim's eyeballs would burst from the immense pressure, and the victim’s face would be covered in disfigured, gaping holes.
Aneurysm, Asphyxiation, Death
The human body is for obvious reasons built to handle the force of gravity from top to bottom, and not their reverse. When hanging upside down blood-pooling in the head will eventually cause brain aneurysm. A more immediate threat however is gradual asphyxiation due to the weight of organs pressing down on the lungs and diaphragm. Assuming these two health risks don't happen you will also suffer damage to the eyes as the pressure inside the eyeball doubles when the body is turned upside down, over prolonged periods this can cause permanent damage to the persons vision, possibly even blindness.
