I'm planning a trip in September, which would place me in MAD at around 7:35pm Saturday, night, from Lufthansa flight. And then departing from MAD at 9:35am on Monday.

Based on the arrival/departure time, could anyone suggest what might be good for a quick look around. First time in Spain.

If I read it right, LH arrives/leaves in T1?
I was thinking of just taking a taxi, to make things easier, from airport to Atocha and stay around there for 2 nights (Saturday night/Sunday night), and taxi back to the airport. Any recommendations for a good hotel to stay in? It doesnt have to be 5 star, just a place to crash at night, and safe.

I also read some people talking about how crazy the Madrid airport is. How early should a first timer get there on the day of departure?

Thanks!