Hi, jaemep:

Your fig tree is still alive(since the underbark is green), maybe it is just going through a transition period from dormancy?!
One expert who has about 30+ fig tree growing in Canada told
me not to overwater a fig tree as this can kill it easily!
But still give some water to this little tree to prevent it from drying
out since it is out in the sun!

herman2 is one expert you can ask your questions!

You have to dissect the growing buds ,starting from tip,and go down toward roots.
If the buds are brown,the tree is dead to that point,and the first dissected green bud you find,That is the point where your tree start to be alive,so you Know.
Cut the dead wood now or later it is OK,any time.

Also, this may provide some encouragement:
http://figs4funforum.websitetoolbox.com/post/Hold-on-to-that-dead-fig-tree-6353390

Thanks for all the advice, I have a glimmer of hope now.

Hi, jaemep:

When there should be new growth, a tiny patch of green color peaks through the bud. Just be patient!