It has been pre COVID (May 2019) since our last visit, but we stayed at Sleepn' Atocha. Funky hotel near the Prado and about a 12 block walk to Plaza Mayor. It is also very close to the rail station (Atocha). If you stay downtown recommend you get tickets back to airport the day before your flight. This rail station gets real busy. Or you can get a taxi - $$$
Blog Link: Hotel Sleepn Atocha - Arrival Madrid
Hotel Link: Sleepn' Atocha
Also remember when you arrive the hotel room may not be available until 3 pm... or so
However, if you are just there for a day after travelling the night before, I would look at the Marriott by the Airport. A beautiful facility, with a great pool and restaurants and bars. Great place to kill a day and rest up for your adventure. Best of all they have a shuttle to the airport. Yes, it does not have the charm of downtown, but it sure makes an "Easy Button" out of your short stay. The Marriott has a plethora of rooms and I am sure one will be available.
Blog Link: Marriot by Airport - Departing Madrid
Hotel Link: Marriott Airport Madrid
Madrid airport is huge and the last time we did it (before COVID) it was 2 hrs and 15 minutes from the hotel shuttle to arriving at our gate. Lots of Walking!
So if you have a noon flight and our downtown not sure what time you would have to leave.
Let us know what you decide