One of the most common questions I get from advanced benders is how to deal with the pressure on the neck and throat in chest stand.
To a certain extent, that pressure is inevitable. If you look at the the x-ray photo you can see how much of the flexibility for this position comes from the upper back and neck. Also, if your feet are not on the floor then you are supporting most of your weight on your chin and that causes pressure as well. However, there are a few things you can do to make this more bearable.
Stretch your hip flexors. The more flexibility you have in the front of your hips the less work needs to happen in your neck to get your feet to the floor.
Make sure that the front of your shoulders are on the floor and press them down to take some of the pressure off the throat. Do exercises to strengthen the neck and upper back so that they do not feel crushed under the weight of your body and can support you safely.
Always think about extending the chin forwards and lengthening your neck rather than crunching into it. Practice! The more time you spend in the position the easier it will get as your body is able to relax and breathe.
That said, if you have ongoing upper back and neck pain from doing this pose, you may need to take a break and figure out what is going on with your neck as any previous imbalance, weakness, or injury will be exacerbated by hanging out in this crazy position. Good luck!