You just got the aero-bars so of course it is going to take time to get used to them, common sense. The more you do something the easier and more natural it becomes. Your neck muscles will strengthen and adapt. Go for a ride and sit up with your hands on the hoods, drop down onto the drops or aero-bars for a bit until it gets uncomfortable then get back up on the hoods. With time you will spend more and more time down comfortably. Once you can stay down for the length of a 20 or 40 kilo TT you are in good shape.

While riding fast on the drops or aero-bars you do not have to have your head looking straight ahead every second. If the road is smooth and straight ahead for many feet or yards then look down at the line on the road for a little bit, turn your head side to side a few different ways to keep it from getting stiff or cramped. I am in my 60s and I ride down on the drops all the time with no problem. Me on my race-bike;