We are in the rainy NorthWest, so we are pretty up-to-date on the best weatherproofing methods.
You’ll want to seal your oven with an ORGANIC Masonry Sealer to help repel water (and stains) - This is a low-cost, first-step when “weatherproofing your investment on a budget”.
The best way to WEATHERPROOF your oven is to simply cover the oven with a thick, clear piece of plastic sheeting. This can be purchased at most Mom & Pop / local hardware stores - for about $10. It keeps all the moisture out - nice and dry inside.
To the second part of your question - Applying several layers of paint on the stucco is an excellent way to seal a portion of the oven, but keep in mind, as long as there is exposed masonry, there is an opening for moisture. If masonry gets wet, it will act like a sponge and soak-up the water (wicking).
So unless you can afford a custom made cover for your oven (every BrickWood Oven is different - so we don’t offer covers), plastic sheeting / tarp is the least expensive (and very effective) way to weatherproof your oven.