An Agent was able to dodge Neo’s shots from across the rooftop and wasn’t taken down until Trinity shot him at point blank range.
Unfortunately, long range weapons like sniper rifles probably won’t work for John against Neo. As witnessed in The Matrix, distance plays a factor in bullet dodging.
While Neo’s dodging ability was seen to be far from perfect at that point, one could imagine he could dodge a bullet traveling from a distance and would potentially sense it approaching.