Learn Polish: Naukowiec ostrzega nas przed tym samym paradoksem. - The scientist warns us about the exact same paradox.

It is in the instrumental case. The dictionary form is ten sam because the noun it describes, paradoks, is masculine. However, the phrase ostrzegać przed requires the instrumental case, so ten changes to tym and sam changes to samym.

I thought przed meant "in front of" or "before". Why is it translated as "about"?

Prepositions often change meaning when paired with specific verbs. The verb ostrzegać (to warn) always teams up with przed to mean "to warn about" or "to warn against". It is a fixed combination you just need to memorize.

Why do we use nas here instead of another form of "us"?

Nas is the accusative form of my (we). The verb ostrzegać takes a direct object in the accusative case because you are directly warning someone.

Why does the word for paradox end in -em?

The base word is paradoks, which is a masculine noun. Because it follows przed in this context, it must take the instrumental case. For most masculine nouns in the instrumental case, you simply add the -em ending.