Which character are you? When I was playing professional I had a slightly easier time with Chris here. I just ran after the spider and when he would come past me I would repeatedly quick shot it.
The difficulty I had with Piers is that his sniper rifle would somehow miss, even though it should be a one shot quick shot. His assault rifle was always on single shot for me, so you could try switching it to automatic for that section.
And if you know the spider-man is around a corner, waiting for you to chase, you can try to throw a frag grenade off the wall to see if it can hit them, they way they don't flee from you getting too close.