Someone I know who is a native speaker of Spanish and a professor at a university in the United States has complained that when he used the word "penultimate" in class, the students don't know what he's talking about.
The word is perfectly, unimpeachably, 100% correct. It is a bit of an upscale, literary word though. It certainly is a "term of art" in fields like linguistics, where it is used commonly to refer to syllables.
I searched for the word in the Corpus of Contemporary American English, and divided the results by section: spoken, fiction, magazine, newspaper, and academic, and got these results:
So, it is certainly used in all areas of English, with a moderate skew towards higher registers like academic writing. Depending on the intended audience, a writer contemplating using the word "penultimate" should consider that there is a relatively high likelihood that word will not be understood.
Instead of "penultimate", the phrasing in common usage is either "next to last" or "second to last". Both of those should be universally understood.
Interestingly, Google Ngrams shows that the word "penultimate" rose in usage (in published books) steadily from 1800 to about 1990, and since 1990 it has shown a steep dropoff, taking us back to usage levels from the late 1960s. The better-understood terms "second to last" and "next to last" are less common (again, though, in published books—Google Ngrams tells us nothing about spoken usage), with "next to last" having more historical usage but a slow tapering off of usage since 1960 and "second to last" increasing in usage from 1970 on, with both enjoying approximately the same usage today.