c. 1400, "having a sharp point; producing punctures," senses now rare or obsolete, from Medieval Latin punctualis, from Latin punctus "a pricking" (from nasalized form of PIE root *peuk- "to prick").
The meaning "prompt" is recorded by 1670s, from the notion of "exact, precise, insisting on fine points," including the observation of time and the keeping of appointments (c. 1600). Related: Punctually.
word-forming element making abstract nouns from adjectives and meaning "condition or quality of being ______," from Middle English -ite, from Old French -ete (Modern French -ité) and directly from Latin -itatem (nominative -itas), suffix denoting state or condition, composed of -i- (from the stem or else a connective) + the common abstract suffix -tas (see -ty (2)).
Roughly, the word in -ity usually means the quality of being what the adjective describes, or concretely an instance of the quality, or collectively all the instances; & the word in -ism means the disposition, or collectively all those who feel it. [Fowler]
<a href="https://www.etymonline.com/word/punctuality">Etymology of punctuality by etymonline</a>
Harper, D. (n.d.). Etymology of punctuality. Online Etymology Dictionary. Retrieved $(datetime), from https://www.etymonline.com/word/punctuality