Hi, angelalisa:

I would wrap the trunk too as I'd never
want the tree to be exposed to the dry
winter wind! I wouldn't want to have the
risk of damage to the fig tree.
I understand it's a heck of lots of work but the fig tree rewards you with the abundance of fruits.
As it takes years to fruit a lot, I would
certainly make sure to bundle up the tree well.

fignewbies

I would trim the tree to a manageable size then use the burlap first then the tarp. The burlap would help to keep it warmer while the tarp would keep it dry and stop the wind. Cover from the ground all the way to the top. Also with the trimmings you can share them with friends or start new trees for yourself.
Good Luck!

Please show picture of the tree. Thx