The Golden Tulip was easy to get to from Incheon Terminal 2. Just 3 stops away at Unseo station. That's about 15 minutes, depending on when the train picks you up. However, to get to the Golden Tulip to Seoul Station, you're looking at around an 1 hour via the All Stop AREX line. That part wasn't as fun.
Our flight out was at 9 am and breakfast wasn't available until 6:30 am. The coffee shop onsite at 7 am. Luckily, there were some restaurants about 2-3 blocks away that were open at 5 am.
Our biggest issue was the Shuttle reservation for our flight. When we checked in, the person who helped us asked about a shuttle. We said yes please and told them our flight was to be at 9 am. They recommended 6 am. We said that sounds good.
The next morning, when we arrived at the 6 am bus, they asked if we had a ticket, which we were never provided with one. He refused to take us. So we went inside and I told the person at reception about the issue. He told us we didn't have a reservation. This made me confused since the person who checked us in made it seem like we were good to take the 6 am shuttle. Eventually, the person that morning said, it looks like my reservation was for 7 am instead. However, either way, we were not provided with a shuttle bus ticket.
This was likely a miscommunication between all parties involved due to language barriers. Outside of that, the rest of my stay was good and it took us less than an hour to get from the hotel to our gate via the All Stop AREX.