Korea - 12-hour ICN layover and trip into Seoul

We have a 12-hour layover at ICN on Sunday, arriving at 5:15 am from Hanoi on Asiana (economy) and leaving at 5:05 pm on United (business), all on one itinerary. I assume our luggage will be checked through. We would like to go into the city for a few hours--on our own via limousine bus, not on a transit tour. We know it will be a Sunday morning, but we're really just interested in walking around a bit (we have a map and some ideas) and having lunch. I have a couple of logistical questions for those who know the airport...

Any issues with this plan with respect to Korean C&I?

If we want to stop in the Asiana lounge after our inbound flight but before leaving the airport, will we be able to get to and get in the lounge in our arrival concourse? We will not have UA business class boarding passes until later in the day when the UA counter opens (because of upgrade co-pays due), but we are eligible through UA RCC memberships.

We will probably want to store our carry-on bags. From the airport map, it looks like there are baggage storage facilities near the B and M check-in counters. Are these landside--i.e., can we get to them before/after check in at UA but before going through immigration and security? Which one would be closer to UA counters?

We know to leave plenty of time for the return and are thinking that we should be on a bus by 1:30 pm at the latest. Is this reasonable, or should we go a bit earlier?

Thanks for your help.

We would like to go into the city for a few hours--on our own via limousine bus, not on a transit tour.

to get into town swiftly, use the AREX which takes just under 30 mins for the trip into the city center

to get into town swiftly, use the AREX which takes just under 30 mins for the trip into the city center

Thanks for the suggestion. That's what we did and it couldn't have been easier (it actually took 43 minutes). From Seoul station, we used the subway to go on to our destination.

To answer a couple of my questions above...

No issues with Korean C&I. We did not attempt to go to the lounge on arrival--we just went through immigration. The bag storage is right where described on the map. It wasn't exactly cheap; we paid about $18 to store three carry ons for about 6-7 hours. There are lockers at Seoul station not far from where you get off the express train.