The one point in the movie where I thought he was channeling the spirit of Shatner, he managed to do it without it being cheesy. It's the point at the end of the movie where he steps onto the bridge, takes a look around, and exclaims "Bones!"
I got goosebumps at that moment.
I liked that moment too, but there were other moments that really felt more like Shatner for me.
The biggest moment was when Pine's Kirk is in the captain's chair during the Kobayashi Maru, and he orders medical to prepare to receive wounded. His inflection and delivery there is very Shatner-esque. I also found the scenes with Bones when Kirk is "sick" also had some great Shatner moments.
I think if anything, Pine should take that sense of wonder, enthusiasm, and positive energy Shatner infused the role with. The Kirk of the first film didn't have those qualities yet, as he still doubted himself and was still haunted by his father's death. But in the new film, I hope we'll see a Kirk who, by no means as seasoned as the Prime Kirk, is filled with passion and enthusiasm for being "out there" and who exudes the confident energy of command that Shatner breathed into Prime Kirk.