No, nothing like that. 1)The lawsuit is a separate civil process that can't itself directly challenge the suspension. They involve some of the same subject matter but are otherwise wholly unrelated.
So Vilma isnt' sitting home waiting for a decision, the case hasn't really even begun yet. But Goodell is expected to file a motion to dismiss, which is a prelimary motion seeking to end the case before it starts . . . 2) he's got a fairly good argument for that (though Vilma's counterargument isn't so bad itself).
Expect that motion process to last into August (and a decision could come even later than that). 3) If Vilma survives the dismissal request, the case will then go into discovery, when Vilma can then get access to the bounty evidence. 4)If it isn't settled or disposed by motion through the discovery period, it will eventually go to trial (sometime next year to be sure). And that's only if it isn't dismissed on preliminary motion.