Third Person (Omniscient); (with some other styles thrown in for good measure)
On the whole, the omniscient third-person narrator of The Epic of Gilgamesh sticks to pretty basic narration of the "He said, she said; he did, she did" variety. Thus, for the most part, both the narrator and the reader/listener are "outside" the action, looking in at it. For example, the text opens with descriptive lines about Gilgamesh: "He saw the Secret, discovered the Hidden, he brought information of the time before the Flood. He went on a distant journey …" (1.4-6).
You get the idea: it is a lot of description of events from the outside. That said, there are some moments when the narrator actually takes us inside the thoughts of the characters. Our first good taste of this is just after Enkidu has finished his frolicking with Shamhat. After all that loving with the temple-prostitute he realizes he'd like a friend (naturally). But before Enkidu can even tell Shamhat that, the narrator lets us inside his head for just a moment, "Becoming aware of himself, he sought a friend" (1.195).
At times like these, the story switches into the "filter" type of third-person narration, where the characters' brains become like those semi-permeable membranes you learn about in biology class: letting things (like the reader and the narrator) pass through by osmosis.
Finally, at some points characters talk in their own voices about things that happened to them without the narrator interrupting. Sure, it's still in a third-person frame, but it goes on for so long that it might as well be a first-person monologue—like when Utanapishtim tells Gilgamesh about how he and his wife built a ship and escaped the Flood. This monologue, told entirely in Utanapishtim's voice, goes on for more than 200 lines. Take a breath, why don't you!