Take a look at the city's official bus website for from-to points at
http://www.emtmadrid.es/mapaweb/emt.html . The website takes a little getting used to, but if you click one of the bus graphics on the Plaza de la Cibeles a window will pop-up to show you the BUS STOP LOCATION as well as the bus numbers which stop there. The stop at which the airport shuttle bus stops is at/near stop number 69. At this number 69 bus stop, it says bus number 1, 2, 9, 15, 20, 51, 52, 53, 74, 146, 202, 203, and some nighttime buses, too. If you click on those individual numbers, one-by-one, the map will show you the route of that particular bus number. I see you can get bus number 1 (direction: Cristo del Rey), number 2 (direction: Reina Victoria), 74 (direction: Pintor-Rosales), 146 (direction: Callao), and 202 (direction: Reina Victoria). All these buses will stop on Gran Via, JUST PAST the Gran Via metro station (and Calle de Montera). You'll get off, CROSS the Gran Via heading south, and south on the Calle de la Salud, one block, to the Plaza del Carmen.
http://www.metromadrid.es/ says T1 to Gran Via metro station takes about 43 minutes and involves making 2 changes, 3 total metro lines. The down side of this is the walking more to make those line changes. Whereas taking the express will drop you essentially at the same place you'll get the next city bus to (near) your destination on Gran Vía.
If it's raining (doubtful!!), I'd definitely take the metro to get your near your place without having to wait on Plaza de Cibeles for your connecting city bus, potentially in the rain if there are a lot of people waiting in line. Rain also causes slow-downs and potential back-ups on the highway artery entering/exiting the city.
Saludos, MadridMan