I would book this hotel for the night before you leave, if you need to leave early in the morning, and then I would book downtown for all of the other nights. The airport is not near the center of the city, and you will miss out a lot on being away from the center.
To answer your first question, there is a train that runs from the airport to downtown. To get it you must take the hotel shuttle to the airport, catch a bus (I think it's terminal B) to the train station, then get on the train. The good news is that it is free (or was in April). The bad news is that the whole thing including waiting can easily end up taking an hour or more each away. Look for the RENFE bus to the train station, then it is all easy.
So stay in the city center all but the last night, stay at the Tryp Aeropuerto the last night, and your trip will be much nicer.
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