Suppose a cell is placed in a solution with a high concentration of potassium and no sodium. how would the cellular sodium-potassium pump function in this environment?

The sodium-potassium pump of cell function is to take 2 potassium atoms into cells and dump the 3 sodium outside cells. The pump is also needed ATP as energy to make it work. That is why the pump sometimes referred as Na3/K2 ATPase pump.
The pump cannot do these things separately. That means if there are much potassium outside cells but there is no sodium, the pump will catch and keep the potassium then wait until they find sodium inside cells. If there is no sodium inside cells, the pump will not work even if there is much potassium outside.