You can do it but it will involve some prep work. Id advise taking the image into Illustrator and tracing the lines. This can be done manually, or with Image Trace. Id reccommend by hand as you will have more control. Keep in mind that the direction the lines will be drawn is dictated by the first vertex drawn, so if you want a line to draw from top to bottom, start drawing in in AI that way. You can change the first Vertex in AE but it will save you the bother.
id recommend drawing projections of the object on separate layers. Then, import this layered file into AE as a comp. convert the layers to shape layers, add your strokes (narrower at the “back”, fatter at the “front”).
Add the Trim Paths operator to the Shapes and then animate the End value feom 0 to 100%. Select Sequential as the mode to make them draw on one after the other.
alternatives include the Stroke effect on masks or Write on with positional data.
There’s also an autotrace function in AE that you could try but Illustrator offera more control.