How to water-protect my Arduino?

In a nutshell, you want to treat the Arduino and all other electronic components as you would treat a line voltage electrical system. You want to have a weather-resistant case for the Arduino with as few access ports as possible, and you may want to go so far as to plug those access ports with silicone once you have the required wires in place. The easiest way to protect it from the actual watering system is simply to have the electronics at a higher elevation than the water source, while observing all the previous precautions I mentioned.

Firstly, there are commercially available products like this one.

The easiest thing to do is to get one of those airtight boxes for food storage and place the Arduino in there. If you want any wires to come out, cut a hole and use hot glue¹ or some waterproofing substance (like plumber's epoxy). Silicone waterproof nuts work as well.

^{1. While this will work for the device you have in mind, note that hot glue seals may not work well when completely submerged.}

I used to install equipment onto Communications Towers that would have to be exposed 24/7 to all the elements (including getting covered in ICE).

I typically would use NEMA rated enclosure boxes for this, because the thick hard plastic is almost indestructable. but you can drill into it for Cable Glands or mounting easily enough.(make sure to seal anything you drill)

I use Cable Glands to seal cable entries, and depending on the direction, drip loops should be used.

you could also opt for something with Ingress Protection ratings, (something like IP67-IP69+), check out http://www.enclosurecompany.com/ip-ratings-explained.php for more information on enclosure IP ratings.