First off, I assume this is a poly-plastic drink cooler, not one of those $3.00 styrofoam jobs that say "MILLER" on the side, right? LOL
The odor is from mfg process and storage so a simple initial airing out should suffice. Bleach solution should only be used if you're experiencing a mold problem and need to start over.
If you air it out for a day and still have the odor, put 1/4c baking soda (not BAKING POWDER!) in a gallon of lukewarm water, sponge it down the sides a few times (with a clean washcloth) leave it closed for an hour, then drain it and rinse it with clean water a couple of times, then dry it with a clean towel and leave it open until dry.
I would NOT put nails in my coolidor. I just put my boxes on the bottom, bundles on top, and a desk-organizer plastic "IN" tray for the loose ones. Everytime I get a box with a spanish cedar sheet, I toss it down the sides.