The clues from the text are fascinating, but since so much about the Sumerian culture that produced Gilgameshis still poorly understood, we can only make conjectures about what a king in Uruk in the third millennium B.C. was supposed to do and say. The first thing we learn about Gilgamesh (who was, in fact, an historical personage according to the Sumerian King-List) is that he was "called god and man (Mason 15) ", and that he and Enkidu became human together (this is from the Herbert Mason translation, Book I, and this is called in other versions "Gilgamesh, Enkidu, and the Netherworld). Gilgamesh, who was obviously a great and powerful person, was a tyrant, because he claimed an "old birthright,/The privilege of sleeping with their brides/Before the husbands were permitted" (ibid). So, we know by negative evidence that this custom of ius primae noctiswas allowed in the past (before Gilgamesh, because he receives the rights as a birthright) but is now considered tyrannical by the people of his day. So from this we can see that the idea of tyranny (excessive rights for or oppression by a ruler) was known to the Sumerians, and abhorred.
This book continues, saying that Gilgamesh further impinged on what was considered the proper behavior of a ruler:
Sometimes he pushed his people half to death
With work rebuilding Uruk's walls,
And then without an explanation let
The walls go unattended and decay,
And left his people dreaming of the past
And longing for a change.
They had grown tired of his contradictions
And his callous ways. (16)
So, again from negative evidence we learn that a king was not supposed to overwork his subjects, nor was he supposed to let infrastructure (such as defensive walls) become dilapidated. He is supposed to be consistent in his ruling, and not be callous toward his people. Gilgamesh is not the model of a good ruler at the start of this poem.
Many times in this poem, however, Gilgamesh is called god-like. This is considered a good thing in a king, however, and the Sumerians considered kingship a form of, or at least a gift from, divinity.
At the end of the epic, when Gilgamesh confronts Utnapmishtim, he learns the secret to eternal life. Utnapmishtim tells him there there is nothing permanent in this world. Though Gilgamesh goes back to Uruk without obtaining eternal life, he is certainly wiser. From this last part of the tale we might draw the conclusion that Sumerians wanted their kings to be wise, and to understand the impermanence of life.
Mason, Herbert, trans, Gilgamesh: A Verse Narrative. New York: Mentor, 1970