"Cause" is the direct object, so a stripped down sentence would be: "Heaven will crown a cause". "Just" is an adjective that modifies "cause". It is not only A cause, it is a GOOD one, a JUST one. "So" is an adverb that modifies the adjective "just". It acts to emphasize "just". It sort of turns "just" into "very just", but not exactly. It means something like "just, as this cause is just".

"crown" -- a verb that literally means "to place a crown on someone's head" but means something like "to reward" in this context "just" -- in this context, it's an adjective that means "fair" or "rightful" "cause" -- in this context, it's a noun that means "undertaking" (it's the same "cause" as in "It is a good cause" or "I'm fighting for this cause because I believe in it.") This sentence is very poetic and is likely to be found in written text rather than spoken language (unless it's a very formal speech). "With" in "Heaven will crown with success" means that Heaven will reward the cause with success. "So just a cause" means that the cause is especially fair, good, rightful, etc. The short answer is that this sentence means "The cause is so good/rightful/fair that Heaven will make it successful."