I have where someone else has already done something similar in nature and I am attempting to "reverse engineer" the process.
...and what do these older files tell you about the steps you should take? It would be a good starting-off point.
Furthermore, if you are using After Effects as a novice, you should understand that the simple possession of old project files don't help a lot: there will be modifications to be undertaken, and you will need to know the After Effects Basics at the very least; as you might imagine, there is a good deal of arcane knowledge that is assumed in using the application. You can get that basic knowledge here.
The basics are to use a stroke as a track matte to reveal a path. Search for animating a signature tutorials. Try setting a black solid over your drawing, then change the blend mode to Screen so that the layer is no longer visible. Now use the pen tool to draw over your drawing in the order you want to have the strokes appear. Do not close the paths. Now add Stroke, on Transparent to your black solid and make the width wide enough to cover all of your drawing. Now set the layer to be a track matte for your drawing and animate the end position of the stroke. That should do it.
The example file you posted looks like stroke is not applied over a drawing, it looks like the only thing on the layer. IOW, the paths are the lines.