"Hij spreekt een beetje Nederlands."
Translation:He speaks a bit of Dutch.
Correct! The exact reason is that "a little" should be modifying Dutch. In other words, the amount of Dutch he speaks is a little bit.
If you switch it, it changes the meaning. In RonanSill's example, 'a little' modifies how he speaks, not the amount of Dutch. In other words, the amount that he speaks Dutch is only a little