Electronic – Trouble Identifying Electrolytic Capacitor

I'll be an outlier here and guess that it's a (blown) 0.5A "pico fuse" style fuse, and I'd be pretty sure if it's the first thing connected to the mains wire.

Break it apart you'll quickly see what its made of (and can easily rule out inductor, if there's no copper wire inside). A fuse and a resistor may not be easy to distinguish.

If it is a fuse, and if it's blown, there's a significant chance other stuff is blowed up good. Poke around and check the power semiconductors and diodes.

Edit: For those who have little experience with Asian manufacturing, here's an example of the type of fuse, mostly supplied by Chinese manufacturers:

The 'cement' finish in the OP's photo (rather than the smooth lacquer you'd expect on an inductor or resistor) is another not-so-subtle clue as to the functionality.

You are correct. Generally speaking, capacitors must not be subjected to voltages higher than what they are specified for. In practice, one always chooses a capacitor with voltage rating somewhat in excess of the highest voltage the capacitor might be exposed to. For example, I would choose a 63V capacitor for a circuit running at 45V.

Your circuit is running at plus minus 20V. A capacitor voltage rating of 25V is marginal, so I would choose a capacitor voltage rating at 35V or more. Your 50V capacitor will be just fine.