Hi Sean,
I looked at your previous post and let me first day sorry. Somehow I read that your arrival would be 6 PM! This would make it impossible to reach it the same day but 6 AM gives you ample time. Sorry I really have no idea how I missed that.
It is still quite a trip but here's how to go about it. First take the train to Kanazawa like this. You should arrive at Kanazawa between 12-13.00 as it is about 4H by train. From Kanazawa station you can take an express bus to Shirakawago.which takes an other 75-90 minutes.
buses only leave at 8.45 12.35 16.00 so you'd probably have to use the 16.00 bus however it would be possible to make the trip. Be sure to reserve your bus tickets the moment that you come to Kanazawa as the bus is by reservation only. Also you would have to stay the night in Shirakawago as there won't be any return buses the same day.
I hope this helps and good luck with your trip!
Daniel
PS: I'd still recommend at least considering renting a car from Kanazawa, simply because it allows you to come and go whenever you please.