I have some anodized aluminum bottles that have faded, but not uniformly. Before painting them, should I simply apply a coat of primer, or do I need to sand them? If I sand them, should I sand off the entire anodized layer?

I hope they are not green color, that would lesson the value of the them.

Why not have them re-anodized.

No you should not sand aluminum, you will destroy the metal polished surface, then you would have to re-polish them, and they would make the bottles even thinner thickness of metal.

You can try paint remover, not the quick stuff, but the thick kind. It should remove the coating. Then clean with Lacquer thinner.

Have you looked into powder coating?

Powder coating might add too much thickness depending on the tolerances involved. But it is an option.

I agree completely!! I have even discovered that adding a label can affect the tolerances. I was redoing a set of Multiplying Bottles and just gluing on a label that I'd printed on copy paper cause a jam, where adding an inkslide decal label did not.

I would highly recommend completely removing the old finish before refinishing.

Google: removing anodized finish from aluminum

You will find information.

But as Richard pointed out, powder coating is an option. It would depend on what the prop actually is. With some, extra thickness would not be an issue. With others, it might.

I have altered several sets of bottles. They were cheap because the paint went off. I sanded them and used spray cans from a car repair shop. I found some very dark green. Now they really look like glass bottles.