I just got the "Schutzenschnur" award, is there is a certain way to put it on the ASU's?

Take the jacket to Alterations and have a button sewn centered under right epaulette and another button under the lower portion of the lapel

AR 670-1 chapter 29, section 19, paragraph c, it states:
29-19 c. The German Marksmanship Award (Schuetzenschnur) is authorized for wear only by enlisted personnel. Officers may accept, but may not wear the Schuetzenschnur. If authorized, personnel wear the award on the right side of the uniform coat, with the upper portion attached under the center of the shoulder loop, and the bottom portion attached under the lapel to a button mounted specifically for wear of this award.

So, beyond misspelling Schützenschnur, AR 670-1 tells us that the top part of the cord under the center of your shoulder loop, and the bottom part under the lapel to something that you have to customize yourself. I have seen people use safety pins (no one noticed); I have also seen people sew very small buttons under there themselves (that worked too).

Congratulations! When I wore mine, I would kinda curl the cord under the bottom of my nameplate. Works but the sewn button method is better!