I have had this problem as well. I would be fine for about 2 hours but with longer times in the saddle severe neck pain would come, also for me it was worse when going up steep hills (raising my head more).
The ultimate solution is to raise your handlebars- at first try at least level with the seat, that should give you some improvement but in my case I had to go about an inch higher. Stem risers can be purchased just for this reason but you're steering tube may have a few spacers that you can remove to get some improvement. Flipping your stem over may give you relief and costs nothing.
Also if your helmet has a visor get rid of that because it forces you to bend your neck to get it out of the line of site(that helped me a little too).
todays emphasis on "racing" bikes leads people to believe they need a low stem/handlebar combination. This is entirely different than people rode years ago. Check out rivendell sit and Sheldon Brown for more extensive discussion . People who tour or take longer rides are probably more prone to this than people who take shorter rides, also as we get older our necks are less flexible and people who once had no problems suddenly notice this . Good luck.