My question is simple, What to do after you transplanted your rooted cutting?
The rooted cutting is now placed in a container with a nice mix that provides excelent water drainage.
My question is, do I put the cutting in full shade for a while (if so, how much time) or do I put the cutting in a place that gets partial sun?

To answer your question it would help if you provided your location and zone.  Where I am it would make no sense to put such a fig outdoors.  If you are in a place where the temps at least get up into the 70's at least during the day and now lower than the 50's at night you might try the outdoors in partial shade and see how it does.  Protect the plant from rain so that you can control how much moisture it gets.

What size container?

If the cutting has good roots, it should be fine. If not, you're gonna have to baby it. Does it have leaves? You need to harden the plant off if you're gonna put it in full sun. Watch it carefully. Find the balance between not letting it dry out and not watering it too much.

I had similar cuttings some with more and some with less Root growth, small clear cups with 100% vermiculite soaked in water after cuttings been placed in.
Shake off excess water, place in styrofoam box like I have with some clear wrap over it. Over time it's wise to find a way to let the box breathe...
On the cheap side I changed a small bowl of heated water daily and after 4 or 5 days i,could see root growth on the edges of the cup
Note I found cuttings excellerated root growth once I placed it in the vermiculite and kept the humidity and temperature up.