Will this rust? If so, is there a product that can be sprayed on to rust-proof it?

Nothing (other than esoteric chemistry) protects carbon steel from rust as well as good tin galvanizing. You can add coatings, you can wash and dry it each time you use it, those efforts can help, but you've already reached diminishing returns when you buy galvanized steel because it works so well.

If properly cared for the pail should last a lifetime

All metal will eventually rust if left in wet or moist conditions. However, a product called "Corrosion X" can be purchased and used to prevent rust on metals.