Can I use a polarized cord and plug to rewire a non-polarized clock?
You're absolutely right.

Since the original plug is non-polarized, the clock doesn't care which wire is live and which is neutral.  With the polarized plug, live is always going to be on one wire, and since the clock doesn't care which one, neither should you.

The only thing to worry about is that the new wire is the same gauge or thicker so that you can be sure that it will safely carry the current that the clock requires.