Yes, definitely take the train to the city centre, where there's much to do, rather than stay in the airport. Since you are in T1/T2, the easiest option is probably to take the subway, rather than go to T4 to take the commuter train (there is a landside arrivals lounge in T4 where you can pay to take a shower, but I think it will probably take too much time). There are left luggage facilities at MAD so your carry on is not a problem.
The lounge in T2 is in Priority Pass and you can usually buy day passes for it (the shower is probably extra, although I am not sure). They will only let you enter three hours before the flight.