Songlines by Bruce Chatwin — Chatwin's classic 1987 attempt to understand Aboriginal mapping of the land through walking and song. The Centre is the book's spiritual core; it has dated in places but is still the most evocative outsider's account of country like this.
Tracks by Robyn Davidson — Davidson's memoir of walking 2,700 kilometres across the desert with four camels in 1977, much of it through country south and west of Uluru. Honest about the loneliness, the heat, and what it costs to do something genuinely difficult.
Why Warriors Lie Down and Die by Richard Trudgen — Not specifically about Uluru, but the best introduction to why two-way communication between settler Australians and First Nations communities is so difficult and so important. Worth reading before the cultural centre.