A fluorescent fixture on the header is a very fine way to light a closet. You may want to think about adding a manual switch instead of switching to the door. Having to make sure the door is fully closed may get to be a hassle. Many of the fluorescent light fixtures have rocker switches or pull chains on them, if you do not want to go through with the trouble of having a wall switch installed.

The clearance requirements for fluorescent lamps is much more generous than using incandescent lamps. I think they are a minimum of 6" from the nearest point of storage space. Assuming you have a closet shelf, draw an imaginary light vertically from the edge of the shelf to the ceiling. Your fluorescent fixture must be at least 6" away from this line. As always, for code issues you should check with a qualified electrician.