Answer
The $Na^+$-$K^+$ pump required ATP to operate. It would cease to function when the cell dies, because there is no more available ATP.
Osmosis doesn't require any ATP and thus will still occur even after cell death.
Work Step by Step
Active transport needs ATP to operate. ATP is created by living cells. If a cell dies, it doesn't produce any more ATP.
The $Na^+$-$K^+$ pump is an active transport. It requires ATP and thus a living cell to operate.
Osmosis works passively by flowing down the gradient and doesn't need either ATP or a living cell. You can actually do osmosis experiments with non-living substances.
