So how old is Godzilla in this movie?

He's the last of his kind. 

Sadly there is no way of knowing Godzilla's true age.

Just to clear things up, the toy descriptions say that he's from a time when radiation was about 10 times as plentiful as it is today. They don't say anything about a time before dinosaurs. I'm not sure where that came from, but I've checked the toy line ups and I don't see anything about dinosaurs.

Anyway, Godzilla's true age is best left as a mystery. Every answer we discover demystifies him that much further and takes away from his appeal and mystique.

According to Godzilla: Awakening, he is a reptilian creature from the Permian period. Therefore, he is, at the very least, 250 million years old.

He's millions of years old. But whether he's the last of his kind if unknown. Presumably he is, but maybe there could be more.