Q:

Cases of attributive adjectives describing units of time (Month, weekdays, etc)

Treffen wir uns nächsten Freitag in demselben Cafe?
Is the reason for "nächsten" taking the weak dative ending because it is the short versrion of "Am nächsten Freitag"?

A:

No, the grammatical case used with a preposition and the case used without a preposition are two entirely different things.
If a preposition is used, the case is basically determined by the preposition. With temporal an, dative is used.
Without a preposition, accusative is used. There is also a temporal genitive but it is no longer used today except for some relicts with temporal adverbs like "nachts".
The nächsten Freitag in your example is not a weak dative but a strong accusative.

