I've only checked in for late departure flights from City Airport Terminal, so take this with a grain of salt, but IC Grand Parnas did have a shuttle van which can run you over to CAT. Dunno what the operating hours were/are; you might want to check with the front desk. But if you have to, it's not a long walk, provided your baggage isn't an issue. I'm sure other hotels around CAT have similar service.
I was recently instructed by Korean Air that I should consider checking in to CAT 3.5 hours before my flight, but again, I've not checked in for a flight before 2 pm via CAT. But this was through their "Voice of the Customer" Web form, and probably answered by someone here in the USA, not someone on the ground in the ROK.
Take it for what it's worth, but
the earliest you can go through immigration at CAT is 6:00 a.m.