My husband works in an office environment and wears long-sleeved shirts every day. When the cuffs start getting frayed but the rest of the shirt is still fine I make a short-sleeved shirt by cutting off the bottom part of the sleeve and hemming it up on my sewing machine. He wears these altered shirts in the summer months.
A suggestion--go a little longer with the sleeve than you would think to. I know from experience it is really easy to make them too short and then you've wasted the whole shirt! Compare the sleeve to a commercial short sleeve shirt that fits well, before you cut.