Learn French: La ponctualité est essentielle ici.

In French, general concepts and abstract nouns like ponctualité require a definite article (le, la, or les). English typically drops the article when talking about general ideas.

Is there a reliable way to know the gender of a word like ponctualité?

Yes! The vast majority of abstract French nouns ending in -té (which usually correspond to English words ending in "-ty") are feminine. Notice that unlike many other feminine words, they typically do not take a final silent -e.

Why is essentielle spelled with a double L and an E at the end?

It is agreeing in gender with the feminine noun la ponctualité. The masculine singular form is essentiel. To make adjectives ending in -el feminine, you double the -l and add an -e.

Since this lesson is about être censé, how could I rephrase this idea to say "One is supposed to be punctual here"?

You could say On est censé être ponctuel ici. The expression être censé (to be supposed to) is conjugated to match the subject, and then followed directly by an infinitive verb—in this case, être (to be).