I remeber you could get a one day fishing licence in the Agriculture Ministery who allows you to fish in public allowed places in the allowed time.
I don't know if it's still the same.
I've seen people fishing in damns in Madrid, but not fly river fishing.
If I guess something I'll tell you later.

Now are the Comunidades Autónomas who give the fishing licences, so I guess you get the licence only to fish in that region. For Madrid you have to get a full year licence but it's something like 11 €
Have a look here
http://gestiona.madrid.org/psga_gestiona...rocedimiento=91

You can see here where and when you can fish in Madrid:
http://medioambiente.madrid.org/areastematicas/biodiversidad/caza_pesca/cotospesca.pdf

another link
http://medioambiente.madrid.org/areastematicas/biodiversidad/caza_pesca.html