Learn German: Fleiß ist sehr wichtig für den Erfolg. - Diligence is very important for success.

Traits and virtues like Fleiß (diligence), Mut (courage), or Geduld (patience) are typically used without an article in German. Erfolg takes the definite article here because it refers to a general but concrete outcome rather than an internal trait, though saying für Erfolg without the article would also be grammatically acceptable.

In spoken German, can I shorten für den to something like fürn?

Yes, in very casual spoken German, you will often hear für den squished together into fürn (like fürn Erfolg). However, this is strictly colloquial and should never be used in standard writing.

How does the noun Fleiß relate to vocabulary I might have learned earlier?

You likely learned the adjective fleißig (hard-working or diligent) early in your German studies. Fleiß is simply the base noun form, meaning "diligence" or "hard work."