Try on T-shirt. Mark button placement at approximately an inch below elbow on the side of each sleeve.
Slide the zipper inside the sleeve, wrong side of the zipper against the wrong side of the sleeve. Align the bottom end of the zipper with the mark on the sleeve. Sew a few hand stitches to secure the end of the zipper to the sleeve, being careful not to catch the other side of the sleeve in the stitching.
Sew the button to the mark, stitching through the sleeve and ends of zipper.
Roll up sleeve. Pull the zipper from the inside of the sleeve around to the outside. Unzip the zipper a few inches, place over the button. Zip to close around button.