"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.