This won't be much helpful considering the part you need to do but you did right wearing the quest costume when entering GT and going to the first portal. My information about the quest, however, is outdated since I last did it in 2010. I just wonder is the other quest meant to be finished when in actual GT raid. When you get the first portal you can summon Aidenus and turn-in quest and take new one for the boss. I can't remember was that before or after the 1st boss though. I hope someone else that has quested GT more recently can advice you better and further.