Yes. Once the poly is completely dried it will not affect your cigars at all.
You might have a problem with the "cider"

but as far as cedar is concerned, the humidity alone isn't going to hurt since its like living in a humid city to begin with (like me).
Honestly its a lot like refurbishing a bathroom. A good primer and paint would work fine. You could also use green board, which is a water resistant drywall, it tends to sag on ceilings however so the joists need to be closer together.
If it were me, I'd just use plain old drywall, a good primer, and a couple of coats of a mildew resistant paint, then your cedar. Be sure to allow for some expansion in the cedar as well when you install it since the humidity will cause it to grow a little.
I would install wonderboard, just like installing ceramic tile in a bathroom. You can also get a better bond with the cement board.
My $0.02
spiffy
ps Welcome to the forum!
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