I've built a couple of these over the last few years and I think I caulked the first one and painted afterwards and on the second one the other way around. No particular reason.
I really don't think you need to get too anal over this as the combination of the paint and the caulk will stop the water from getting to the wood. Just make sure you caulk sufficiently and get a couple of good coats of paint on the wood and you should be just fine.
I used about 3/4 inch plywood so there is no flex in it, and as I said, it only has to channel the water to the drain.
Remember we're not going over Niagara Falls in it.