Did Marbury's case ever go to the lower federal courts?
No. William Marbury filed a petition for a writ of mandamus (an
order compelling an official to take - or refrain from taking - a
legal action) with the US Supreme Court, which is the head of the
federal court system.
The Judiciary Act of 1789 assigned original jurisdiction to the
Supreme Court for writs of mandamus against government officials,
which Chief Justice Marshall decided was not the Constitution's
intention. According to Marshall, Marbury's case was not within the
Court's jurisdiction; he would have to file with the lower court
(District Court) for relief.
Marbury never refiled his case.
Marbury v. Madison, 5 US 137 (1803)