Why pick up the car at the airport? Unless you're a seasoned "Madrid driver", the drive into the city and search for the hostal and parking place might be pretty frustrating. I would spend a night at the hostal in Madrid and pick up the car the next day from a central Madrid location.

When arriving in Madrid from the US, we always spend the first night in Madrid. I will not drive the first day because of jet lag. But Madrid is so vibrant and exhilirating that I love wandering and eating my way through that first foggy day. Sitting around inside a hotel room (other than for a BREIF siesta) has never occurred to me. Relax and enjoy it.