First person doesn't really exist in its true form when it comes to movies. We suppose directors can go all out to really create the experience of a character in the film, but after watching Cloverfield, we've had enough shaky hand-cam for a lifetime. The point is that until virtual reality takes over cinema, to have a real first person movie is basically impossible, unless someone's doing on constant voiceover. We're rarely able to experience the exact same world a character in the movie is experiencing.
But Gondry wasn't daunted by the task of essentially simulating a first-person experience. How does he do it, you ask? Well, he doe it by providing viewers with a limited amount of information. When Joel wakes up the morning after the Lacuna procedure, he has no clue what has happened—and neither do we. He goes to Montauk and meets Clem for what both he and we assume is the first time. There's only a brief period near the end of the movie (before Joel plays the tape) when we are aware of what has happened while Joel is not.
By messing with the films chronology, Gondry restricts what we viewers know so that we are never too many steps ahead of Joel.
A cool example of this happens in the scene where Joel looks at himself in Lacuna and realizes: "I'm in my head already, aren't I?" Gondry had given us hints, like when Joel walks out of Barnes & Noble and re-experiences talking to Frank, but this is the first time it becomes clear that we're actually witnessing the procedure take place, and it's the first time we really get a sense of the direction the film is headed.
We should probably mention, by the way, that we literally are in Joel's mind. The film could have just been about Joel's memories—he could have been reminiscing about the past and reflecting on what brought him to this decision to have his memory erased. But instead, we delve right in and actually interact with Joel's past in a very visceral way. It doesn't get much more first person than that.