I am fond of Hostal Inn Madrid, Mesonero Romanos, 10 Planta 4, Madrid City Center, 28013 Madrid, Spain. You can find it in Bookings.
It is not fancy, but it is a fairly-priced hostal, very clean, and right by the center of Madrid. It's a short walking distance from Callao metro station. I get from and to the airport using the metro, although admittedly the route can be a bit convoluted and confused if you have never used Madrid's metro. (Anyone attempting to use the metro, remember that an additional Airport fee must be added to the metro fare - another bit of information that is not at all clearly explained to tourists!)
Anyway, whether you get there by Metro, bus, taxi or Uber, I have found it to be a great place to spend a night or two in Madrid at a reasonable price, albeit definitely not luxury. There are also a couple of small restaurants and bars right on the same alley where the hostel is, that serve local food and used mainly by locals, at a very reasonable price. Far better than the many tourist traps a couple of blocks away.