This sounds very uncomfortable! The skin on the eyelids is thinner and more delicate than the skin on the arms or neck, and this might explain why it burnt so quickly in your case.
The first thing that needs to be done is to get a sense of how serious the sunburn is. The best way to know, in your case, is probably to go and see your regular primary care doctor. Many mild sunburns do not need much in the way of treatment, other than the application of a mild skin moisturizer. On the other hand, more severe sunburns, especially those with blistering, are at high risk of scarring or infection and should be managed with the help of a doctor.
Your doctor will be able to help you figure out quickly whether this is burn requires any special treatment. They'll also be able to give you advice on how to relieve the symptoms of a minor burn. As I mentioned above, mild skin moisturizers may have a cooling effect that relieves some of the discomfort. At the same time, anti-inflammatory medications like ibuprofen or naproxen can help with the pain and redness quite a bit, as long as your doctor says that they are ok for you to take.