You can easily get from Hanoi to Ninh Binh by train or bus - we went a couple of weeks ago: got a taxi to the Giap Bat bus station on Giai Phong, straight onto a bus (20 seater) and it took about 2 hours. It cost 45,000VND each and they run regularly.
Or I'm pretty sure you can get an Open Tour bus there - there are lots of posts about it anyway so have a look.
From Ninh Binh you can get the train down to Hue. The train times aren't ideal as the only ones stopping in Ninh Bing are the SE5 (departs 1440 - arrives 0107), the SE70812 (departs 0812 - arrives 1958) or the TN1 (departs 1236 - older trains). So might be worth looking at the Open Tour bus option.
When we were in Hanoi last year our hotel told us we couldn't get tickets for the train from Ninh Binh to Hue, so you might have to go to the train station in Hanoi and buy them directly. It's cheaper that way anyway.
So definitely do-able with just a little bit of extra effort.