While, within the film, they never actually show him dead, cutting to the shooter and the sound of the gunshot, according to Zack Synder, he is indeed dead.
But why kill off such an iconic DC Comics character? Well it was all in the name of ‘fun’ according to Snyder.
He told Entertainment Weekly: ‘We did it just as this little aside because we had been tracking where we thought the movies were gonna go, and we don’t have room for Jimmy Olsen in our big pantheon of characters, but we can have fun with him right?’
Of course, such deaths often don't stick in comic books or the movies based off of them.
The release of Batman vs Superman: Ultimate Edition includes a scene where the character's CIA handlers observe the death and indicate that their code name is "Talon", a name used for an elite Court of Owls assassin who in the comics has the power to regenerate from injury. This suggests two more possibilities of the fate of this character, namely that
either the real Jimmy Olsen may be alive, unaware that his identity has been stolen, or the character killed is Jimmy Olsen, but is also Talon, and will eventually regenerate from his wounds, possibly after the CIA removed him from the scene.
I personally find the theory farfetched, but I include it for the sake of completeness.