So this rip in the sleeve is a common problem in men's shirts that get a lot of wear, pressure at the elbow and a gradual thinning of the cotton means eventually they just give out. Usually it is a neat rip in a straight line rather than the jagged tear you get from snagging it on something.
Now a lot of people would just throw the shirt in the bin (wasteful!) if the shirt was looking worn or damaged in other areas or was no-longer regular wear then maybe I would break the fabric down to be used as something else, failing that take it to the charity shop. They do get paid for the weight of rags they receive also so don't go thinking things are not good enough to take, just check with your local store.
The shirt is in very good condition every where else, it is still a favourite (Paul Smith shirt found on a sale rack a few years ago) and therefore worth the time.
So what I usually do is just add a fake seam going across the back of the sleeve to hide the frayed, torn bits.
Turn the sleeve inside out and pull it flat...
Sew a straight line of stitching just on the back part of the sleeve tapering it as you start and finish just like you would if you were sewing a dart. When you have done that set your machine on a small zig zag and sew across again to capture all the frayed bits. Then repeat on the other sleeve making sure you sew in exactly the same place, this way your shirt will not look odd.
Now on the good side as you can see this shirt is checked so you lose a little pattern, however this is a close up shot and it is not noticeable when hung up or worn. I think someone would have to know and be having a really close up inspection of your elbow to see it. ( Who does that?)
Another nagging little job done and it didn't take long at all. If you have more damage to shirt sleeves than this you could always consider making them short.
I have also used this similar method on my son's chino's when he fell and made a small hole to the knee (they were new!) I did the same seam across both knees and on that occasion I also top stitched it. It just looked like a design feature and so saved them from being ruined.