well, I also had to serve the army, and 1 month must be a period for training, and it is not allowed using personal stuff during training. in my case, my bro posted my date of enlistment on SNS, so my friends could write a letter on the internet. idk if you can write it cuz you are in the US now but if you know his date of enlistment and his name and age, maybe you can find him on Korean Army training center homepage and write a letter for him.