Eu sunt managerul acestui* restaurant
That's the Romanian equivalent for "I am the manager OF this restaurant". But I agree that most people would rather phrase it the same way you did.
Is it correctly pronounced like an English speaker would do or not?
Romanians seem to pronounce ”manager” the Romanian way, like in the audio: with two clear ”a” (like English ”but”) and with the stress on the last sylllable.