"their namesake" is a reference to the ancient Greek myth of Narcissus. At the end of the story of Narcissus, he dies and a narcissus flower grows in his place. He is the namesake of the flower. Narcissi (flowers) are named after Narcissus (the person). Additionally, "doing poetic justice" to something means doing something that symbolically relates to it. It is best to learn this phrase by example. "that way they do some kind of poetic justice to their namesake's eagerness to be seen" Could be paraphrased as "by blooming first, the narcissi are reminiscent of Narcissus's vanity." The narcissi bloom early as if they want to be seen before other flowers. Narcissus was obsessed with his looks, so the narcissi trying to be seen early makes you think of Narcissus's vanity. In this way, the narcissi do poetic justice to their namesake.

This is the reference to the Greek myth of Narcissus. He was a very handsome young man who fell in love with his own reflection. He would stay next to the water and stare at himself all day. The writer is saying that the flowers, which are named for the beautiful man, are just as vain as the person for whom they are named.

A namesake means a person or thing having the same name as another person or thing. Narcissi are named after Narcissus, because of a similar attitude.