AdjacencyGraph will create graph with your matrix and then you can do TopologicalSort over it.
m = {{0, 1, 1, 1, 1, 1, 1, 1, 1}, {0, 0, 0, 1, 1, 1, 1, 1, 1}, {0, 0,
0, 1, 0, 1, 1, 1, 1}, {0, 0, 0, 0, 0, 1, 1, 1, 1}, {0, 0, 0, 0, 0,
0, 0, 0, 1}, {0, 0, 0, 0, 0, 0, 0, 1, 1}, {0, 0, 0, 0, 0, 0, 0,
1, 1}, {0, 0, 0, 0, 0, 0, 0, 0, 1}, {0, 0, 0, 0, 0, 0, 0, 0, 0}};
g = AdjacencyGraph[m]
g // TopologicalSort
{1, 3, 2, 5, 4, 7, 6, 8, 9}
