Thanks for all the replies. I have a few more questions if you don't mind.
So renting a car is not recommended. Can I take a taxi from the airport and ask him to make a small quick tour on the way to Madrid's center? Or that's not a good idea?
Can I use the Velazquez Lounge on arrival before going to the city? I assume I can, but do I need to go back through security?
I understand it is much better than the Arrivals Lounge, right?
What are the breakfast and lunch hours at the Velazquez? Are they published anywhere?
Thanks again.