Marte was promoted to the Pirates Wednesday night, and he will likely become the Bucs' starting right fielder. He had played well all season for Class AAA Indianapolis, and had been regarded as one of the Pirates' best prospects. The Bucs sent reliever Evan Meek to the minors on Wednesday to make room for him.
Rodriguez will be with the Pirates on Thursday as they play in Rodriguez's former home in Houston, but it's not yet clear when he'll pitch. His last start was on Monday, so it's possible the Pirates could have him start Saturday.
A.J. Burnett will pitch against Dallas Keuchel on Thursday. The Pirates haven't yet announced who will start against Jordan Lyles on Friday, but it will probably be Jeff Karstens. If Rodriguez does indeed start on Saturday (with the Astros' starter still to be announced, since Rodriguez would have pitched that game if he'd stayed with Houston), James McDonald will likely pitch Sunday as the Bucs face Lucas Harrell. (David Todd at Bucs Dugout has a good rundown of how the Pirates will likely handle their rotation for the next few days.)
One the Pirates do formally add Rodriguez to the 25-man roster, they'll have to remove a player, presumably a position player. Rodriguez is already on the Pirates' 40-man roster.