The Bridesmaids and Moone Boy actor admitted that he has "been kind of lucky" to enjoy a gradual rise to stardom.
"The fame thing has happened very gradually, so I've been kind of lucky," he told Closer.
"I don't know how I would have dealt with it if I had been one of the Twilight guys."
However, O'Dowd also admitted to overestimating his level of fame, explaining: "A couple of weeks ago, a tourist stopped and handed me a piece of paper.
"I thought he wanted an autograph. But then I realised it was a map and they were asking me for directions."
O'Dowd married longterm girlfriend Dawn Porter late last month.