I was told by a guy that I should install a 2x4 like this all along my fence on the top rail to help reinforce it to hopefully prevent bowing or something.
Would this be worth the $200 in screws and wood to do it?
When I have built wood fences I always install the 2x4 like that. Take one, prop up one end and stand in the middle and then rotate 90 degrees and repeat, which way is stiffer?
Nobody will be coming to save you, plan accordingly.
Wooden fence is going to warp like a motherfucker regardless how you install the stringers unless you live somewhere that the humidity is virtually constant.
No doubt the long side will support a greater load, but there is nothing to say it won't warp in that dimension.
I always install the top rail as a T, with the top plate horizontal and a vertical 2x4 centered on it. The vertical provides support to secure the tops of the pickets, then that the hardware is covered with a 3/8x1 1/2" ripped piece of cedar.