It seems that Turkmenistan’s national railway Turkmendemiryollari doesn’t have a website, and it is not easy to find information about its train services.
However Samuel Rachdi has done a lot of research, and as a result Turkmendemiryollari timetables can be found at his fahrplancenter.com website.
This includes a timetable for the daily return passenger train on the line built by imperial Russia from Mary to Serhetabat (Kushka), close to the Afghan border.
From Serhetabat the railway continues across the border to the freight terminal at Towraghondi in Afghanistan, but (as far as I know) this section carries only goods traffic. “Two kilometers of the 7-kilometer link lies in the territory of Afghanistan”, according to the Afghan Ministry of Foreign Affairs.