My flight leaves at 6:10 a.m. That means I need to leave around 4 a.m. from the hotel, so I want to be sure the hotel's shuttle is truly a 24-hour operation. Any suggestions?
Have a look at these two ...
http://www.clementbarajashotelmadrid.com/
https://www.tach-hotel.es/en/
Or if you are on a budget ...
http://www.accorhotels.com/gb/hotel-3753-ibis-madrid-aeropuerto-barajas/index.shtml
The airport hilton. It's at the airport and you can walk to your terminal in 10 minutes. They also provide a free shuttle
The Maydrit hotel (cheaper than the Hilton) has a 24 hour airport shuttle. It leaves every half hour. The hotel is close to but not at the airport. Nice room, nice bathroom, quiet.
Clement Barajas did well by us six years ago, with prompt 5am shuttle, coffee & dry edibles in the lobby. Best aspect was that the hotel is in a real neighborhood, so the night before we ate tapas in a friendly low-key local three blocks down alongside the neighborhood park, & bought flight provisions in the mom&pop grocery next door. There is or was a hotel restaurant also. Easy access on the metro (we came from the main Madrid train station), if you're okay with several blocks' walk. Nice quiet & modern room, surprisingly sleek hotel given the unprepossessing neighborhood. We'd use it again.