We do know he's likely to be N, because the script pits him as the voice of multi-directional thought / thinking outside the box vs Arthur who is portrayed as the rock/stability/linear thinker. Their arguments are typical N vs S arguments.
Watch his behavior, and he comes off as an impersonal thinker -- detached.
So we now have NT.
Does he seem to like to control situations?
No, he adjusts to situations as they unfold (and does this with natural ease throughout the movie).
His appearance is a bit neat (which is more J), but it's still relaxed; and his manner is more P.
Once you get him down to INTP vs ENTP, he is far too comfortable in social situations and fits a typical ETP pattern (the ESTPs look similar to him as well), and INTP would just not respond that easily and naturally within the exterior world. You can also see the I/E difference between Ariadne (who is IxTx) and Eames -- Ariadne detaches and analyzes, Eames engages.
Overall the best fit is an ENTP who actually is fairly well-rounded, as he can focus clearly and not shoot off in every direction (and a lack of focus is usually the bane of the young/unnuanced ENTP).
Hope that helps.