I think this is unlikely to be a popped blood vessel in the eye. Although these do occur, the tend to cause a large, reddened spot of discoloration under the clear part of the eye where the blood spills out. It sounds like what you have, on the other hand, is simply prominent blood vessels in the clear part of the eye. These prominent blood vessels could be caused by a number of different conditions leading to inflammation or chronic irritation of the eye. For example, seasonal eye allergies are a common cause of prominent, irritated blood vessels. Other causes might be exposure to physical irritants, like wind or dust. Finally, some chemical exposure can cause eye irritation; this might occur from exposure to chlorine in the eyes from a swimming pool, for example. I would suggest going to see your primary care doctor about this issue. They will be able to examine your eyes and help determine what is going on. If this is irritation due either to a physical or chemical exposure or allergies, your doctor will be able to prescribe medications, which might be either eyedrops or oral medications, in order to reduce the inflammation. Good luck!
1. Tea bath.
Make a tea as usual. Leave it to cool enough. When it is slightly warm, let piece of cotton wet by the tea, then put the cotton on your eyes. Eyes closed of course. 5 min in the morning, 10 min in the evening and for 10 min before the bed. Make a new tea every day.
2. Cold compress.
After each tea bath. Take a cool water. As cold as possible. Let another cotton piece wet by cold water. Put on the eyes. 5 minutes with some intervals, never let too long to have pain. There is also gel-mask for eyes. Check local drug store. You keep that mask in refrigerator and apply it to your eye for 5 to 10 min.
3. Of course sleep well. If you stay too late on computer they increase. I use computer for my living, these things happen only if I sleep less, tired and stay late on computer... Otherwise computer screen is harmless.
Do not press on your eyes, don't massage. They are nothing to do with eyeball pressure!!!
Eye massage can only be done by moving eyes rhythmically.