Broadly speaking, a writ of mandamus is a court order to a government official to do its job properly, typically by performing some nondiscretionary task.
It is commonly sought against officials in the executive branch, where there is not always a clear process for appealing a decision to take or not take some action. In the judiciary, there's nearly always the possibility of an appeal, so mandamus is not particularly commonly granted against a judge.
There is, however, plenty of precedent for writs of mandamus against courts. One example that quickly comes to mind is Richmond Newspapers, Inc. v. Virginia, 448 U.S. 555 (1980). In that case, a trial court barred the press from a murder trial. A newspaper challenged the order closing the courtroom, but the judge refused to let reporters in. The paper sought a writ of mandamus ordering the court to permit coverage of the case, and the U.S. Supreme Court granted the writ.
In this case, the government is arguing that the court abused its discretion by refusing to allow an appeal before letting the case go forward. This isn't really an "out there" legal argument, but some people would think it fairly bold, as it is a bit more aggressive than waiting for an appeal and calls more attention to a supposed error made by the judge who's going to be deciding your case.
When it comes to writs of mandamus sought against courts, I would say that these two types of cases -- access to hearings and denials of appeal -- probably account for the bulk of the action. When it comes to the second category, I suspect that most are denied, because doing this without committing a fatal procedural error can be fairly tricky. If you're interested in more about how mandamus works in this context, there's a pretty decent explainer from Wiggin & Dana.