Hi!
The first one is the most natural of the ones you listed, however, I would say something more like "...after he asked for directions TO the National Palace Museum."
The way you wrote it is not wrong, but directions are usually thought of as a way of getting to a place.
I understand your logic in using "the directions", but in this case I think that phrasing may imply that there is only one way to get to the museum. Again, it's not wrong to use "the directions" here, but it is a less natural way of speaking (at least for American English).