I just stayed at the Courtyard on 30-Dec. I used the pre-pay option and it was around USD$70. I arrived while the shuttle was running, and utilized that. It was on schedule, leaving on the 30.
Hotel was ok. Didn't try the food. Walked up the street to a good kebap place.
My flight left pre-shuttle in the AM as well. They called a taxi for me. It was a short ride and cost TL16 ($9) before tip.
Under any other circumstance I would take the tram into Sultanahmet, but short turn around mileage runs are the exception.
BTW: I flashed my Aegean *A Gold Card for access to the good lines, even though I was on KLM that day. Successful.