Basically your talking about SAG (Surface Action Group) warfare here. Something you could expect to see in a China-Japan confrontation or China-Taiwan confrontation. I think that basically it would depend on who had superior radars/stealth and SAM systems. Fire first and kill more of the enemy's missles than they kill of yours and you win. A battle like this would probably be quite short.
The USN has a big advantage in this area because they have both light, survivable supersonic ASMs (the SM-2) and heavy hitting slower missles (the Harpoon). So they can fire the SM-2 and Harpoons at the same time, SM-2s hit the ship massively messing with the radar and SAM systems, disabling them to deal with the Harpoons which kill the ship.