How do I fill part of a rounded rect shape layer in after effects?

Remove the Fill, add a Gradient Fill. Set one end of the gradient to the first color, the other to the second color, then edit the start and endpoints. There's not much to it. You could even add multiple colors and slide the middle point right up next to the previous color. Check out these screenshots. You can even add multiple gradients and fills then use blend modes to combine them. I'm showing you all modifications to the layer.

That should get you started. 

Actually you can click on fill for the shape (top of the screen) and choose a linear gradient and adjust the start and end points and position of the gradient onto the same point so it looks two tone

Well you could use two rectangles and mask out half the top one. Then select those two shapes and pre compose them then add the drop shadow on that pre comp's layer 🤔 🤔