I haven't been to Korea so I can't compare it with Singapore. However, why are you coming here? I guess that's the big question and maybe people will be able to better answer your questions.
I'd suggest only coming here if you were offered a high-paying job and your moving expenses paid. Cost of living is very high compared to the USA, but I don't know how it is compared to South Korea. Check for online cost of living comparisons internationally.
Apartment costs will vary depending on where you live and what type of place you get and whether you're near a train station, near the water, etc. You usually deal with a private landlord here, as opposed to a property
leasing office. Some landlords are tough.
Also, if you do come here, I hope you have an undying love for the following:
Extreme heat every day of the year
Long lines (long Qs)
Trips to the grocery store that last a good chunk of the day