I know this thread is about neck pain and not the handlebars, but...
Are your hands remotely comfortable on the hoods with those bars? Your brake levers are nearly vertical which is odd. It's my opinion that I'd be searching my local Craigslist for some modern-shaped handlebars. I think the odd shape / curvature of your current bars is preventing you from being able to position the brake levers in a proper way. If you're riding on the drops, can you reach the brake levers?
The following is how I'd expect to see one's brake levers on a modern handlebar. There's plenty of horizontal surface on the tops for resting your hands on the bars or brake hoods and you can also easily reach the brakes from the drops:
