Dead Man Walking is written in first person: it's told from the viewpoint of the author, Sister Helen Prejean. For that reason—surprise, surprise—you mostly get what Prejean knows or sees or hears. When she's waiting to see whether Pat is going to be executed, you're waiting too; you can't know what she doesn't.
The book does occasionally leap about in time. The Sister Helen Prejean in the narrative is a Sister Helen Prejean in the past; the book is written by an older, more knowledgeable Sister, and she'll sometimes slip in some details she didn't know at the time but learned later. For example, when she first meets Howard Marsellus, the chairman of the Pardon Board, she thinks he may pardon Pat. But, she says, "Later Marsellus will serve time in a federal prison for rigging pardons…" (3.151)
Basically, it almost seems as if Sister Helen knows the future. So you could say that the viewpoint is first person semi-omniscient. It's all from Prejean's perspective, but Prejean the writer sometimes knows things you wouldn't expect. Nuns aren't God, obviously, but this Sister manages to be at least a little all-seeing some of the time.