If you will graduate college or university soon,
You have to find a job which is related to your major before 7 month passed from your graduation.
If you find a job, then apply work permit as international graduated program.
It will give you work permit very quickly, less than 2 weeks.
If you get a work permit as international graduated program,
you are automatically skilled worker who is eligible for Alberta PNP.
To apply Alberta then you have to work at least 6 month at that job to apply Alberta PNP.
After receiving Alberta PNP, then apply RN card.
I think this is the shortest, and easiest way to get RN card.

About passport, you should contact to Korean embassy.
It is totally different matter from visa or army things.

About army, I have no idea about it. if you get work permit, you may able to stay in Canada.
But anyway you will return to Korea someday. then, you have to go to army.
I recommend to go to army before you graduate school.

Sorry for bad English