Who does Anthony describe as a 'slight unmeritable man, meet to be sent on errands' and what does he send him to do ?

That would be Lepidus.

Third is his rant against Lepidus. He doesn't consider Lepidus worthy to join the triumvirate, and he probably has a point there. Lepidus was selected to mediate between the egos and enormous ambitions of both Antony and Octavian. But Antony's language to describe him is over-the-top. Lepidus is a "slight unmeritable man,/ Meet to be sent on errands (4.3.12-13)." When Octavian tries to inject a more moderate and appreciative assessment of Lepidus, by saying that Lepidus is a "tried and valiant soldier," Antony whines, "So, is my horse, Octavius, and for that/ I do appoint him store of provender (4.1.29-30)." Lepidus only has derivative thoughts and derivative ambitions: "He must be taught, and train'd, and bid go forth;/ A barren-spirited fellow; one that feeds/ On objects, arts, and imitations (4.1.35-37)." Part of Antony's ruthlessness is his conclusion that Lepidus is worthless.

Lepidus is the man, and he is sent to Caesar's home to get his will.

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LEPIDUS