No those that have suggested otherwise are wrong.
If you ignore all resistance and transonic effects, the bullet fire vertically would reach an altitude of 6.24km.
Whist the one fired horizontally (assuming a shot height of 1.5m obove the ground) would only travel 193.5m before hitting the ground.
basic GCSE Physics: using v²=u²+2as and s=ut+0.5at²
I woudl have thought you'd easily get more than 1 km in altitude, but the problem is actually far more complex than it may appear. Take in to considereation that: 350m/s is above Mach 1 at s.t.p hence oblique and normal shockwaves appear; air is infact not invicid, resulting in skin drag from boundary layer formation and the bullet has a large, turbulent wake inducing realtively large form drag; the atmospheric changes between sea level and your final altitude are enough to not only change the speed of sound, but the flow regieme around the bullet and the resulting pressure gradients and drag forces. You could also consider the change in gravitiational field strength as the bullet increases its altitude.
Long words = confused friend = 5 pints of Guinness.
I owe you one of them.