The hotel I booked in Madrid for New Year's Eve wrote and informed me that taxi service is stopping from 18:30 to 8:00 on January 1st. The taxi is usually 30 Euros, but they can arrange private transport for my husband and me for 60 Euros. Our flight back to the US on January 1st is at 11:15 AM, so we really need to leave the hotel before 8:00. I guess we'll bite the bullet and do it (we are staying in the Salamanca area) but another option would be to change hotels and stay near Atocha for the airport bus. Any advice?
Personally I would book the private transport. Everything will be on a holiday schedule including public transportation. Who needs the stress of getting to the airport. Just budget it into your travels.
Adding to Pab's points:
Are you even sure the airport buses are running? If the taxis aren't running, the bus may not be either.
Even if buses are running, they may take much longer than usual if there are road closures due to New Year's Eve events (or clean up the next morning).
I'd look into other private transfer services to make sure the hotel's price of €60 is not gouging for that time. If it's not, I'd take it.
We stayed at Hotel Europa on Puerta del Sol. It was very easy to get to the airport by train for about 1.4 euro. My husband and I left on different days. He left around 8:30 and had no issues. For me, I needed to get to the airport earlier. I'm not sure if my timing was poor or if the trains don't run until 7:30am, but I did wait about 15 minutes for my train. I don't have a train map handy, but we did need to transfer trains after a stop or two. Your hotel can help you with that, ours did. I believe it took about a half hour total. The train will take you to terminal 1 or 4 (can't remember which, I think it is in my trip report for Spain) If you are not at the terminal you need, go out the door to the median where you will find a bus that goes between the terminals (I believe it was lime green). My flight was at 10:30am and I had lots of extra time. Often hotels seem to push taxis. In Europe, I've always used public transportation from the airport to my hotel. So easy and cheap!
If you do change to a hotel by Atocha, the train is even easier! There is no transfer. I'm not sure how much the airport bus costs but it is hard to beat 1.4 euros for the train! (In Madrid, we always used the train vs. the metro. The metro is more expensive and it appears to be much more confusing)
The above points are good. You need to find out the schedule for public transportation for the 1st. Not that it helps you, but I was in Barcelona over the New Years holiday and their public transportation ran as normal (and was very necessary since the celebrating people were making good use of the metro) From a public policy standpoint, I would think there would be an effort to keep public transportation open for people enjoying New Year's (and probably not fit to drive)