On the third day of my road bike riding (never had a road bike before), I went for a 4 hours ride. Then that afternoon and the next day, my shoulders were in pain. Actually, not so much my shoulder, but the muscle between the neck and the shoulder...

Now, I had the bike fitted to my body measurements. I'm thinking is it because my riding position isn't correct, such as maybe I'm putting too much of my weight into my arms. When I was riding mountain bikes, my body was a lot more up-right. This new position is kind of, well, "new" to me. Is it something that I just need to get use to (since it was only my 3rd day of riding). Or should I get the store to re-measure myself?

More likely that you just overdid it, going for that long a ride that soon. Or, you may have ridden with too much weight on your arms, as you suggested. Some newbies also ride with their arms too stiff, which will also give you neck/shoulder/arm muscle problems, especially on longer rides.

Next time you ride, be conscious of moving your hands from the bars to the drops and out to the hoods every so often, which should help cut down on the arm and shoulder soreness. Also try to make sure your arms and shoulders are relaxed, not stiff, especially when you're "stretched out" on the bike.

Four hours on a road bike on day 3 is really pushing it. Where were you keeping your hands during your ride? Down in the drops, up on the brake hoods, on the bar tops, etc.? Everybody has a "favorite" position, but you do need to move around a bit. Also try not to actually lean on the bars; there should be some weight on your hands, but you shouldn't be carrying your body weight there.

Try relaxing your neck and shoulders when riding.
I see a lot of folks ride with their neck, shoulders, and arms tensed up.