It's a judge's duty to decide disputes brought to them, but when people say that judges uphold the rule of law, does it mean that it only applies in the sense where judges were to deliver judgment in an impartial manner?
The Wikipedia article Rule of law favorably quotes the Encyclopedia Britannica defining the rule as:
the mechanism, process, institution, practice, or norm that supports the equality of all citizens before the law, secures a nonarbitrary form of government, and more generally prevents the arbitrary use of power.
In general the rule of law requires that written laws, as interpreted by the courts, be applied to all, and there there be no special exemptions that apply only to particular persons, nor special laws that only affect some people or groups. The rule is a statement of an ideal which is not always achieved.
"when people say that judges uphold the rule of law" that means that the judge applies the appropriate previously enacted law, and does not make a decision based on the judge's own personal view of what the law ought to be. This includes the rule of Stare decesis that things once decided shall normally remain decided, that existing rules are not arbitrarily reversed or altered by a court.
This would include the judge not deciding a case or issue on the basis of personal bias or prejudice, but it means more than that. It means that the law should not,change depend on what judge presides over a case. Again, this is an ideal not always achieved.
I argue that when people say such things, they mean that the judge must uphold the spirit and purpose of the Law -- things which might transcend the Law itself, like morality, history, or inalienable rights.
In America, this is codified in documents and understandings that underpin and surround the Law, but were never formerly defined in it -- like the the history of Law, the Declaration of Independence, and the Pledge of Allegiance. These are pivotal and defining momements of America's definition of itself, but have never truly been integrated as the letter of the Law, yet contain what many still hold has the Rule of Law -- the history and documentation that will be used to understand the purpose of the Law and the Constitution.