The easiest way to answer this, I think, is to ask yourself whether or not there's any point in guns on fighter jets. Certainly every airforce that has fighter jets opts for guns.
Aiming depends upon relative speeds, not absolute speeds - so if you are chasing your opponent, and your opponent is foolish enough to run in a straight line, only air resistance and wake will make a difference. In a vacuum, it doesn't matter what your velocity is.
Likewise the average speed of a bullet is 1,700 mph, so the difference between 1,500mph and 1,700 mph is not that great. As far as bullets go there isn't much difference between running at 200mph and standing still.
So even if you lose the ability to have tight turns (acceleration would be dangerous) at high speed, there's still a lot of purpose in using guns.
However, if you are in an atmosphere and you can run at the same speed as a bullet (1,700 mph) and you can deal with the sound barrier somehow, then there are all sorts of difficulties with a gun, especially if you are shooting at the same direction that you are running - the huge air pressure would cause the bullet to lose speed rapidly and it would become an obstacle to you rather than a problem for your opponent.
For guns to work at that velocity your bullets would need to have their own propulsion - e.g. propelled transonic flechettes - ie, missiles...