In practical terms: LO handling agent has a (mostly) manned ticket desk at Terminal 2A. There is a small LO sign at the furthest ticket desk. I guess they will have to rebook you onto the next LX flight if LO flight from ICN is late. There is also a LO station office and a F/A base office (for Hungarian F/A's that operate the ICN flights) at the mezzanine level. These offices are officially not for public access but maybe would be able to help, if the ticket desk is not manned. I do not recall if there are any LO signs but these are 2 offices pretty close to the post office.

Good luck!

PS. The BUD-ICN-BUD flights do not follow all LO standards - for example there is no port (and maybe some other booze) available in C. Food-wise full standards are followed (if memory serves).