I think the solution depends on how you assume the destribution of population in a cell. Do you assume population of a cell is uniformly distributed in entire area of the cell, or concentrated at the center point of the cell?
More specifically, which one is your preferable population that should be added from the cell to the buffer polygon, in the screenshot? Assuming the cell value (population) is 100.
100, since the cell intersects the buffer
30, since 30% of cell area overlays the buffer * I think this is @nielsgerrits’s suggestion.
0, since the center point of the cell is outside the buffer (100, if the center point was inside the buffer)
blue: a raster cell whose value (population) is 100 red: a polygon (buffer of road) instersecting the cell black: center point of the cell