I found myself in the same situation as you about 1 year ago. My flight from Ukraine arrived into DME around 1230am. Too late for the Express Train. I negotiated a taxi fair with a random guy after clearing immigration. This was probably a mistake since it was unmarked car, but i arrived safely.
One warning. They call these men the "Taxi Mafia" as they prey on foreign tourists. I asked the hotel what the cost of a taxi was before i had arrived. They told me $25-30 USD. The guy at the airport kept showing me this book that had a price converted of roughly $120 USD. I laughed and walked away. He kept chasing me until we agreed on a price of $30.