The site Korea4Expats says the following re: reporting a change in passport:
All foreign residents must report changes in the particulars of their Alien Registration Card (ARC) within fourteen (14) days of the [occurrence] of such changes, including:
- Passport changes, including passport number, date of issuance, or expiration date changes
- Address change (even if only change of unit in same building)
- Name, civil status, nationality changes
- Detail changes (including name change) of organisation for D1, D2 and D4 visa holders
You can go to your Gu-office or to your local immigration office. D8 visa holders can also go to KOTRA or to the Seoul Global Center (where there are immigration officials).
Failure to report can result in the imposition of a fine, the amount of which is determined on a 'case-by-case' basis.
I don't know how authoritative Korea4Expats is, but it's been a helpful site in the past, so I probably shouldn't ignore its wisdom in this instance.
The good news is that I don't have to go to the dreaded Mokdong Immigration to take care of this: I can just visit the local Gangnam District Office (gu-cheong) to take care of everything. The lady at the foreigners' desk already knows me (and likely dreads me; I visited her three times to deal with my previous problem), and I've never seen that many foreigners lined up there at 9AM. So: quick service, in and out. I hit the local office tomorrow morning.