I would like to reconstruct a simple bitmap to vector, so I have put a grid and a layer on top of it.
Is it possible to e.g. use the fill tool, to just fill out one of the grid squares at a time?
Or is there a better way?
Grid is not a vector object, therefore you can't fill it.
The best way to do what you want is drawing the shape you want to fill using the Bezier line tool with "snap to grid" option on. (As I can see on your screenshot this option is enabled for you).
Then your drawing will fit the grid as you place points.
The final step: fill your shape with the color you want, if necessary remove stroke.