Good to see that you are starting out well with your running. I'm a university instructor in Seoul and I've seen many of my students head off for their military service over the years, many to the ROKMC - did not know the running standards were that fast.
Since you are new to running (but you seem to be making good progress) maybe don't worry too much about 1ks or 400m intervals yet - just focus on building up your fitness to running 40min at a time. This was something that helped me when I started running and after long injuries: start with 5min easy then 5 faster, repeat 4 times = 40min. Start with 30min total if you need to. The next week or 2 drop to 4min easy, 6min faster 4 times, and build up from there to 40 min continuous over the next 1.5 to 2 months. Don't worry about your pace - just pace yourself on each run to make your last "interval" the fastest, and in the beginning it's ok to walk the easy periods (although a slow jog is better.) You are young, you will build up fitness quickly. If you are getting some pain as you run every day, maybe back off and run every other day, or 2 days on 1 day off the first couple weeks, then 3 days/1 day off - you need some time for your body to get used to running. You may not need to run every day, but aim for at least 5 days a week.
When you've reached a level of fitness where 40min of running is getting easier, after about 6-8weeks, then maybe 2 or 3 days a week start trying something called fartlek (it's a Finnish word I think) which is doing periods of faster running by feel, without really worrying about time, and then letting yourself recover while getting back up to a good pace. By feel is the key, but as you get fitter you will naturally be able to do more faster running, and still get back up to pace more easily. Your fast running could be a few minutes at a time, with as much recovery as you need. On days when you're not doing fartlek you can throw in shorter sprints (not full out, but comfortably faster, for say 30s - or try counting breaths while you run faster, 20 breaths, going up to 30 or 40) repeated several times in the middle or at the end of your run. If you do this regularly in month 3, running 5-6 days a week, you should be fine for your goal, you're young with a lot of upside.
You don't need to time your intervals or anything, but in month 3 if you want to check your progress you can do timed intervals maybe 2 days a week - you can make these sessions something like 2min faster/1min jog 8-10times, or3min/2min 6 times. And if you want to test yourself, try sessions like 10min/5min/5min faster with maybe 3min slower jog between - again you dont need to worry about your goal pace, but yoy may find you'll get closer or under 4min/km.
Maybe the best thing that can keep you motivated is to see if your school has a recreational running club, or better yet see if there are still things like the Nike Training Run - running clubs from many schools used to go to these things. Was good fun, when I was able to jog a couple years ago.
I envy you. I wish I could start running again today, doing exactly what I've suggested above...but... My knees won't let me run any more. Once you've started and stay consistent, you'll improve your fitness faster than you realize. Good luck.