We would love to see some parts of the Takayama trail but one of our group is unable to take long walks.
Could we get an idea by travelling to some places by rail? What posts/stations are along the way, if any?

I'm not sure what trail, exactly, you mean -- but no one needs to walk any great distance. There's plenty to do and see in Takayama.
http://www.japan-guide.com/e/e5900.html

I believe that you mean this trail:

http://www.japan-guide.com/e/e5906.html

My younger son and I set out on the trail, but we turned back after a little while, since it seemed not so exciting. It is quite a long trail, so if one member of your group is not up to it, then I would suggest just strolling around the town. There are wood carvers, historic houses, good food, and lots more.

Oh, and I also recommend a day trip to Shirakawa-go. The houses there are not too spread out, and in some cases, you can climb into the lofts, but it is purely optional. You take the bus there and I would allot maybe 3/4 of a day for the whole adventure.

One can also climb into and around Gasshō-zukuri -- "prayer-hands" houses -- at Takayama's Hida no Sato -- no need to go to Shirakawa-go unless you want to do so.
http://www.japan-guide.com/e/e5901.html

There's no rail in Takayama other than the JR line.

There is a bus. The map at the link that CaliforniaLady gave shows the route as a grey line going east from the bus center and labeled "to Okuhida". I don't know where (or if) the bus stops near the trail, only saying that a bus does go to Okuhida. You could take a taxi. Am not sure it is worth the trip given that the main part of the town is easily walkable (and Higashiyama means eastern hills/mountains, so not sure how far you would get).