I got my hands on a fig cutting and am trying my hand at propagating a new tree for the first time. I'm doing the method where you lay it horizontally and cover it with damp soil in the hopes of getting several new trees out of one cutting. I've got roots forming on the bottom end, nothing in the middle, and what looks like a baby fig pushing through the soil at the top end. Do I need to pinch off the baby fig or should I leave it?

Welcome to the forum. Some people say pinch the fig. It gives the plant more chance to develop.
I just let them grow and do what they want. In most cases the fig ends up dropping or drying up.

leave it and lid for SOME days with a plastic cup

--VS

I doubt you will get several trees from one cutting. Even the longest ones I have done this way (18 or 20 inches) did not root all up and down the stick. What you are more liable to get is a good start on a bush form of fig, 2-4 trunks coming up. But I doubt there will be enough roots, and spaced enough, to chop into separate trees.

But keep it well watered and in maybe 1/2 day shade if possible. You will be pleased with the results.

I'd wait till I was 100% sure it is a fig and not a branch bud.  If it's a fig I'd cut it off.  It will slow the growth of everything else.  I'd also stand it up but that's just me  :)

Welcome.
I agree with Bob (rcantor).