Or will I have to buy MoP and some game time?
As Shammoz said, expansions do not come with game time. This can be found in the Authentication Key FAQ:
What type of authentication keys are associated with World of Warcraft and what do they do?
World of Warcraft Battlechest keys come with all retail copies of World of Warcraft and now include access to The Burning Crusade and Wrath of the Lich King content. After creating your account, you have 30 days of play included before your first subscription payment is required. The Cataclysm Expansion keys are available to be used to upgrade a single World of Warcraft Battlechest account to the Cataclysm expansion. The Mists of Pandaria Expansion keys are available to be used to upgrade a single Cataclysm account to the Mists of Pandaria expansion. The Mists of Pandaria and Cataclysm expansions do not include additional game play time.