It is possible to assemble bikes outside the airport I think. You can't ride over the bridge from the airport island legally, but you could take the ferry.
So you want to take the ferry, get off near Wolmido and ride to the Ara canal? Or ride through Incheon and into Seoul?
Riding up to the end of the Ara canal from Wolmido is possible, but if you're straight off a flight, with newly reassembled bikes and hitting Korean roads with city traffic, I'd advise caution.
Also, Montbell may be a bit expensive. If you need camping gear, I'd go to Jeongno 4 Ga (the intersection east of the big Jeongno 3 Ga station) and hit the market there, as it's a huge cluster of camping/outdoor places. So much cooler than a regular Montbell outet as well, you can see Dongdaemun and have a look through all the markets there, get ground mungbean pancakes (so much nicer than they sound in English) for lunch in the famous market by the camping market and experience Seoul.
Edit: Seoul is bicycle friendly, but on the Han river paths mainly. There are many designated bike paths on pavements, but no one pays attention and they're crowded. Patience is required, but there is a whole lot of great infrastructure in the Han River park.
Bring locks, but theft isn't all that common. You don't have to worry too much about things being stolen; like Japan, Korea is well-known for safety.
Also, check out infrastructure here:
https://map.naver.com/v5/?c=14150710...,10,0,0,0,dhef
All in Korean, but the Naver App for smartphones is in English. Here, though, you can see the maroon lines of bike paths. Dotted lines are secondary paths on pavements etc. Don't trust dotted lines; some are excellent open paths, others are afterthoughts. For streetview, click the inverted teardrop icon to the top right of the page.