The hotel is in a nice area, but not in the touristic centre. However, public transportation between the hotel and the centre is very good. The fastest way is to take the subway: take the line 10 at Santiago Bernabéu station and get off at Gran Vía station, and you will be in the middle of everything in just 10-15 minutes. Line 10 is new, safe and fast, and the trains frequency varies from 2 to 6 minutes, depending on the time.
From the airport you can also use the subway, as robert suggested. The line number is the 8, to Nuevos Ministerios. Nuevos Ministerios is quite close to your hotel, so you can either walk from there or transfer to line 10 for just one stop, to Santiago Bernabéu station.
If you want to see a map of Madrid subway, visit www.metromadrid.es. The subway opens at 6:30am and closes at 1:30/2am. The website of the public buses is www.emtmadrid.es. They work 24 hours (from 12pm to 6am, there are special services). If you prefer taxis, in Madrid there are plenty (not as much as in NY!) and are not very expensive (about the same price than yours and much less than other European cities ones).