This is not medical advice.
Sometimes we have an injury or condition that prevents us from rotating our arm overhead. This problem is mostly at the shoulder. I have several articles describing why this happens, or why your shoulder hurts in the first place.
Provided you have full range of motion, and are not restricted, this next progression may help you lift your arm overhead. At this point it is about strength and leverage. If you have any restrictions, are tight, or have a frozen shoulder, that is another issue that needs to be addressed first, preferably in person with a therapist.
When we start to lift our arm when standing, gravity is against you the entire lift, and leverage is very poor. If you start by lying down with a bent elbow, leverage is improved, and gravity is only against you for half the lift. If you start to go past 90 degrees, gravity will assist you in lifting your arm overhead.
An injured or weak shoulder has poor mechanics and leverage. You want to start lying flat, as in the pictures, and practice this maneuver many times until it is easy, and without pain. I have a weight in the pictures, however start with no weight until you can progress with success.
This is called a lawn chair progression. Over the weeks, as you get stronger, start to raise the incline you are sitting in. The first picture at the top of the article shows a gradual incline.
As this becomes easier, raise the incline some more, as shown below.
Sometimes a towel slide along the door jamb can help you accomplish this. Just be very conscious to not “hike” your shoulder, or cheat too much to get it up there. If you are having to struggle or cheat too much, it’s better to go back to the incline progression.
There should be no pain, and preferably you bring this to a professional for guidance.