You can take the EA bus from the airport till Luis de Morales stop (it's where the Sevilla FC stadium is, there is a big mall as well) It will then be 3 mins to the hotel, same block
To go to the center, you can use the metro stop that is close to your door, 3 stops to Puerta de Jerez, or take bus 32 that will take you to Campana (depending on where in the center you want to be). Walking is about 20 mins walk to Plaza de España/University area and a bit more to Plaza Nueva, all pleasant walk.
As for where the center is: in Spain, in almost any city, center is close or around the Cathedral and/or Plaza Mayor. Historically, center is also called to the neighborhoods that were inside the walls of the ancient city. If you take this definition of center, Seville's city center is second largest in Europe after Istanbul, so center in Seville could mean lots of places.