Hi Karlireland,
Welcome!!! I'm thrilled to see another person here. Ireland is a beautiful country and I'm happy to see it represented here.
I'm not sure if you have to write USA as the country or not, but you may need to. Simply write the address on the letter/package exactly as you received it from AS. Regarding what country to write in, ask your postal clerk. I'm sure he/she will be more than happy to assist you. APO AE addresses are in the United States and that's how the military gets them and ships them forward to the soldier(s).
You will need a customs form and I'm fairly certain your local post office will supply them to you as our post offices here in the States do.
Again, welcome! Happy mailing. By the way, if you'd like to increase your chance of hearing from your soldier, you may want to include a self-addressed, stamped postcard. They get "free mail" when they mail something to someone here in the States, but I'm sure not how Ireland's post office works that out. It also helps if you put your email address on your letter so if the soldier has internet access, they may drop you a note that way. Good luck!

Sharon