Artemidorus and Metellus Cimber have entirely different suits to present to Caesar. Artemidorus has a warning letter, which he reads aloud to himself at the beginning of Act 2, Scene 3. It begins: "Caesar, beware of Brutus. Take heed of Cassius. Come not near Casca. Have an eye to Cinna. Trust not Trebonius. Mark well Metellus Cimber...." At the beginning of Act 3, Scene 1, he urges Caesar to read his letter immediately, saying, "O Caesar, read mine first, for mine's a suit / That touches Caesar nearer." Caesar rejects it, saying, "What touches us ourself shall be last served."
Later in Act 3, Scene 1, Metellus Cimber begs Caesar to rescind the decree by which Cimber's brother Publius Cimber was banished from Rome. Caesar in a long speech beginning "I must prevent thee, Cimber, / These couchings and these lowly courtesies / Might fire the blood of ordinary men, And turn preordinance and first decree / Into the law of children..." Cimber is one of the conspirators and is only getting close to Caesar in order to stab him. All the conspirators, including Brutus and Cassius, are only urging Caesar to grant clemency to Cimber's brother in order that they may crowd around Caesar and then attack him. Caesar keeps refusing to reconsider. Suddenly they all attack him and he dies in the long, pivotal Act 3, Scene 1.
Artemidorus and Cimber have entirely different reasons for approaching Caesar. Only Artemidorus has a paper, and it is a letter rather than a petition. Caesar is already thoroughly familiar with Cimber's problem and hardly gives the man a chance to speak.