You're free to hope or think anything you want, but 60+ years of medical study tells me that Kirk had between zero to almost no warning of anything wrong until he was shot and lost consciousness between instantly to almost instantly. With no blood to the brain (0 - 0.5 seconds), it is like a stroke and the brain loses consciousness immediately.
Based on the medial data and visual record, I give Kirk between 0 - 1 seconds after being shot before he lost consciousness, but between the injury to his throat, neck and spine, there may have been too much trauma during that time for him to be awake, alert or thinking.