13 hour flight - something to help him sleep?
Talk to your doctor & see if she recommends any meds.
Buy some baby antihistamines (OTC) and try them at home and see if they help make our BB sleepy.
Bring along your baby carrier. When my kids were babies I walked up and down or stood in the back area of the plane to jiggle them and keep them quiet. I joked about having "walked across the pacific" several times.
When you arrive, try to get BB out into the sunshine to get onto the new schedule.
With a 12 hour time difference, it usually took my babies & toddlers about 5 days to get on the new schedule.
Plan some "shifts" w/ your partner and your family. When I visited Boston alone w/ my 8-month-old son, it really helped that my younger sister helped look after the BB in the wee hours when I was exhausted.
Will you be holding your babe or will he have his own seat and carseat? I've done it both ways with my children (Japan to the states and back several times and Japan to the states to Belgium), and the carseat is so much easier on the plane. I've only drugged one of my children once (I was flying alone with an active 3 year old and a 10 month old DS who'd just had surgery a few days before. . .I gave my DD benadryl while on the longest flight). . .the rest of the times I haven't needed to. I think the sounds of the plane might work like a sound machine. . .it is a lot of white noise. My DS at 10 months wasn't a problem. I nursed him during take off and landing (and any other time he needed or wanted it), I had some new toys I bought just for the flights, I took a sippy cup to pour his drinks into (not that he was drinking much then, but it was fun to play with), and he was pretty easy. Jetlag is tough with little ones. We can force ourselves to stay awake during the day and sleep at night, but a baby can't do that as easily. It usually takes my DS at least a week to get over jetlag. . .even now at 6 years old. Like a pp said, get out in the sun as much as possible during the days. . .it will ease the transition to the new time. Also, when traveling I always take extra clothes for the kids and an extra shirt for me because I know something will be spilled on me before we land. I also like to take some child friendly healthy snacks on the plane. I take hand sanitizing wipes when traveling (it's one of the only times I use them) and a washcloth or two in my bag. I think it's easier to sling than use a stroller and I try to pack light since you'll have to go through customs and pick up and re-check your bags both in Korea and in the states when you come back.