We ripped the guts from a propane fridge, built a box with studs, lined the studs with underlayment plywood, spray foamed everything, and surfaced everything with tile.
Use 2x2s or even 1x2s instead of 2x4s whenever possible. Cut styrofoam scabs to fill the difference. R-value of wood is about 1 per inch, so you loose a ton of temp through wood that touches both outside and inside your cooler.