Learn French: J'ai une mauvaise note à cet examen. - I have a bad grade on this exam.

You are right that most French adjectives come after the noun! However, a small group of common adjectives describing Beauty, Age, Number, Goodness, and Size (often remembered by the acronym BANGS) go before the noun. Since mauvaise (bad) describes "Goodness" (or in this case, badness!), it is placed in front of note.

Why use cet instead of ce for "this" exam?

Examen is a masculine word, so normally you would use ce for "this". But because examen starts with a vowel, saying ce examen would sound choppy. To make pronunciation smoother, French changes ce to cet before masculine words starting with a vowel or silent 'h'.

The English translation says "on this exam", so why use à instead of sur?

Prepositions rarely translate perfectly word-for-word between languages! While sur means "on" in a physical sense (like a book resting on a desk), French uses à to talk about getting a grade "on" a test. You are essentially saying you have a bad grade "at" this exam.

Does note just mean a school grade?