To answer your question, oral argument is scheduled for November 1. Briefs are due in August.
To answer a question not presented by the OP, "this case" is just the request for a preliminary injunction. The District Court still has to decide the case on the merits. And at least one of the "deficiencies" cited in the decision pertaining to the injunction might be corrected. That is the part of the law that is read as mandating 100% of arrests to be submitted for immigration check, the second sentence of Section 2(B): "Any person who is arrested shall have the person's immigration status determined before the person is released."
Lord help us if Bolton gets to take another whack at it.