New Years Eve in Madrid may very well be the busiest single night of the entire year - and finding accommodation is difficult without significant planning and anticipation. In other words, try to make your reservations no later than, say, October 1st. Many "hostales" won't take reservation farther in advance than that - and if you try, they are usually "programmed" to say they're full when, in fact, they just don't take reservations that far in advance. Hotels, on the other hand, take reservations at nearly any time.
Price ARE higher for New Years Eve and the days surrounding it, it's a peak season. Prices probably won't be double but they may be 25-35% higher than normal.
Now, where to stay. Anywhere near the Puerta del Sol is usually booked first because that's where all the action is. You might consider someplace on the edge of the Old City Center like around Paseo del Prado or Calle de Toledo. But, if you CAN get something close to Gran Via, Puerta del Sol/Calle Arenal, Opera, then take it.
Don't wait too late!
Saludos, MadridMan
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