Hi hablas espanol?
I think you will love Madrid.
For a hotel I suggest you get a Bancotel talonario http://www.bancotel.es/esp/funcionamiento.htm
This will get you a room in a 3 to 4 star hotel for 50euros +iva
By all means go to the Plaza Mayor but don't sit on the terraces their tourist traps, you can get the same much cheaper else where.
For spotting I suggest if you have a car to drive to Paracuellos del Jarama great view of the entire airport. Also drive to IB
industrial area, from the roadway you can get a good view of the aircraft approaching.
Hope you have a Good time while your here.