The date and account given by Eisenstein is inaccurate. To the contrary, I found a published statement from Rav Asch regarding this matter in the Yudishe Gazeten of August 19, 1881 (see below). The account relates how some weeks prior, a rav from New York finagled a heter from Rav Yitzchak Elchanan Spektor (1817-1896) to travel on Shabbos on the elevated trains. Rav Spektor wrote Rav Asch a letter relaying the sequence of events. He stated that this New York rav had warned him in a letter that if a heter to travel with the locomotives on Shabbos was not found the fundamentals of yiddishkeit could disintegrate. Additionally, this rav mentioned that Rav Yoseph Saul Nathanson (1810-1875) allowed carrying in New York City on Shabbos. Therefore, Rav Spektor wrote, taking the above into account, he allowed traveling on the train providing that the tickets were bought prior to Shabbos, the traveling was within the techum Shabbos, and that the traveling was on behalf of a mitzvah. Rav Asch wrote that after he had informed Rav Spektor that as this would not cause the collapse of yiddishkeit and there was no heter to carry in New York, there is no heter on Shabbos to travel on the trains at all. Consequently, it is clear that Eisenstein is mistaken that Rav Asch would allow traveling on Shabbos on the train as long as it was not done overtly and it was on behalf of a mitzvah.
Facsimile of the Yudishe Gazeten of August 19, 1881 discussing the permissibility of riding the train on Shabbos in New York.