I have had some similar issues, but mainly in my mid to lower back. I found that if I climbed while holding on to the hoods (praying mantis position,) I would end up pulling up on the hoods to counteract the pedaling down force. This put my back to under go those forces, and fatigued my muscles quickly.

I got mainly used to this over many long and steep climbs, doing back exercises in the gym helped also. To relieve this I would sit more up right, with my hands on the tops of the bar.