Seoul drivers can be nuts, kind of hilly and you'd have to cross a very busy long bridge (Wonhyodaegyo) to get back to the hotel from Hongdae or Shinchon (sounds like your kind of area). Yeouido is a business, finance area with no cool neighborhoods, ok food and not many boutique coffee shops. It's going to be a long ride everyday. Biking to Sinsa and the Garosugil would be nuts. Just use the subway. Cheap, clean and takes you almost everywhere.