The first bit's reasonably easy, I think: Cleo is just imagining what Antony might be doing on the other side of the world. And, thinking that he might be on his horse, she envies the horse - as he has Antony sitting on him!
She then tells the horse to do its job bravely: as the person it moves is the "demi-Atlas" of the earth (Atlas was the Titan who held the world up on his back!). Antony is not only holding up the world, but he's also the helmet ("burgonet") and armour ("arm") of all men.
Cleo then imagines Antony speaking, and asking for her by her nickname ("my serpent..."). But this thought is bad and good ("delicious poison") for Cleo then thinks about herself in a critical way. She is, she says, black ("Phoebus" is the sun - the sun has pinched her skin lovingly ("amorous") and made her dark-skinned. And time has given her deep wrinkles.
Cleopatra then imagines back in time, to when Julius Caesar was alive. When he was here - above the ground - not buried - then Cleo was food fit for a king ("morsel for a monarch"), and just from looking on her face, great Pompey would stand wide-eyed ("make his eyes grow") looking at her face. He would fix his gaze there ("anchor his aspect") and practically die (means "orgasm" as well as "die") by looking on her: his life.