I went for a work trip last year and did 2 days riding at the end, so I'm by no means familiar with the area and others might be able to give better advice.
In short: Yes, it's a great area for cycling!
I hired a bike from these guys:
http://www.rutaspangea.com/alquiler-bicis-carretera/
I went for the carbon bike which was quite old, but did the job ok, except the brakes could have been better on the long downhills, but as I was taking it easy I didn't mind stopping occasionally to give my hands a rest and let the rims cool off. They were very friendly in the shop and spoke ok English and also answered all my questions and gave route advice by email.
I stayed in the Holiday Inn near the bike hire place and I picked up the bike in the afternoon and did a short ride along the river to the palace and back. A very nice traffic free route, although shared with pedestrians it wasn't a problem as it was very wide. No problem to keep the bike in the storage room in the Holiday Inn.
The next day I followed the cycle route out of Madrid, again along the river which was fine and then through some suburbs where the route wasn't always great (a bit like the NCN - not always the best sign-posting or the most direct route), but it was easy enough to follow and quite safe. There is then a cycle route next to the motorway all the way to the mountains. It wasn't great always cycling next to the motorway, but easy to follow and it was only around 60-90 mins next to the motorway. It was a very busy cycle path and felt a bit like a sportive with many other groups of cyclists. There was also a nice cycle cafe - perfect for a first stop half way along the motorway. Once you hit the mountains, it's pretty much like the Alps or Pyrenees. Long uphills and long downhills, small villages with shops and cafes.
I'd pre-booked this B+B here:
http://www.villa-concha.com/
Probably the nicest B+B I've ever stayed at :-) Lovely food, lovely rooms. Shame it was too cold for the pool.
The second day was drizzly/ rainy and didn't want to risk the mountains with the crappy brakes so I took a more direct route back, again following the motorway cycle path and the same route back into Madrid. The sun came out when I got back to Madrid so I did a few loops around the local park/ nature reserve which is near the palace.
It was a shame I didn't have time to explore more - it's a very nice area.
My Strava tracks with pics are here:
Day 1:
https://www.strava.com/activities/412163031
Day 2:
https://www.strava.com/activities/412162940
I hope that gives you some ideas and let me know if you need any more info!