So, it seems to exist a correlation between the RPM of the projectile and its velocity over wind's influence that guarantee a linear calculation over the RPM of the projectile.
If the target is behind a rock, the soldier uses a laser rangefinder built into the gun to get the distance to the rock, and then inputs into the gun "the target is 2m behind the front of the rock". So, unless the explosion radius is small, the soldier has to make sure the bullet explodes behind whatever is between the rifle and the target.
Given that this is basically an ultra-fast artillery device, the tech in this gun more likely represents a natural evolution of applied external ballistics (http://en.wikipedia.org/wiki/External_ballistics) - incidentally this is a purpose for which the ENIAC was designed. I bet the magic has more to do with being able to collect and characterize environmental data through sensors and plug these meaningfully into Ballistic models.
Suppose the bullet leaves the muzzle rotating at 300,000 RPM (quite reasonable for a rifle). That is 5kHz. It's easily within the capabilities of a Hall Effect sensor to measure a 5kHz rotation and so I'd make the very centre of the bullet contain a free rotating magnet. The rifle itself sets the magnet spinning at some known rate opposite to the rifling. When the bullet leaves the rifle it is able to measure the time between detections of the magnet and given the known rifling, the known magnet rotation speed it would be able to calculate rotations.
Alternatively, the inner magnet could be spinning in the same direction as the rifling but faster and detect complete rotations of the bullet. Either way I think Hall Effect + spinning magnet at bullet core would work.
Also, 5kHz is well within the operating range of a micro controller so this wouldn't need much computer power. Lastly the article says that the weapon is accurate at 500m and that the shrapnel kills everyone within a few meters. Also says that the soldier estimates the target distance.
That sounds like there's plenty of room for the measurement to be not too accurate. Suppose 'several meters' is 5 then you can get away with an error of 1% or so.