A stiff neck has at some time probably afflicted all riders that I know which is not surprising when you consider how long you remain in a static position on the bike. I have suffered terribly over the last few years, to the extent that I have considered giving up road biking and going back to mountian biking where the constant movement and more upright position didn't cause as much pain.
As others have said the best way to go is get a professional opinion on the fit of your bike as everything else is trial and error and could compound the issue. I know this because in the absence of a bike fitter where I live I tried just about everything with mixed results. There were 3 things that worked for me in varying degrees. Getting a shorter stem, to push you up a bit so you do not have to flex your neck quite as much when looking up. Getting a rather unsightly and heavy steerer tube extender, essentially lifting the handlebars up 2 inches, and doing specific stretching excersizes whilst riding to keep the blood flowing in your neck.
However the final solution for me was to go with an endurance type geometry (new bike) which fits and my neck pain has all but disappeared.
Im not suggesting you get a new bike which is significantly more than a new stem

but having a proper bike fit.