You can’t control a layer’s in point with an expression, but you might be able to fake it in some situations. You could precomp your shape layer, turn on time remapping, add a slider, and add this time remapping expression:
s = effect("Slider Control")("Slider");
time - s
You might also want to add this opacity expression:
Sure, instead of time remapping the precomped layer, you could attached each animated property to a time offset slider. You’d probably still need to also tie it to the layer’s opacity (unless, as in your example, it animates in from off screen).