Yes you can check-in at New ministerios provided certain conditions are met:
- you are flying with iberia/air nostrum, spaniar, air europa or alitalia.
- Lugage con only be cheched in presented with more than two hours advance (excluded cases where special inspection needed).
When is it really helpful?
- Business traveling, light luggage.
- Securing seats. You can go there and do advance check-in (even the day before)
- National/EU traveling
To answer your last question, the system of going directly to the gate is not an option. Instead e-ticketing combined with check-in automated kiosks are really extended and a lot better option if you are business travelling.
Some info on the topic (spanish only)
http://www.aena.es/aeropuertos/barajas/nministeriosfacturacion.htm