The Millenium Hilton is pretty old and it's on top of a hill with nothing much around the hotel itself. Also if you are arriving at ICN prebook a taxi or take one of the limos because the ICN taxi mafia can be pretty nasty.
If your departing flight is on an airline that can check in at Seoul Station or CALT in Samsung-Dong check in there and get your passports stamped so you can use the fast track at ICN. The lines at ICN in the late morning and evening can be stupendously long because most flights leave at those times. And yes the time saved by using fast track will probably offset the time traveling to the airport.
For food, avoiding Myongdong is a good idea. Gwangjang Market or Jongro is OK, Yeonnam and Sangsu for the latest trends and Cheongdam/Garosugil for expensive and snobbish.
Oh BTW April is prime time for the yellow dust blowing in from China. Come prepared.