The french language indeed has the same idiomatic expression, with a slight modification from your suggestion :

J’eus à peine le temps de sauter dans un train que …

The choice on using either sur or dans depends on the vehicule :

Monter dans un(e) train / voiture / avion

Monter sur un(e) moto / vélo

The rule of thumb is whether the vehicule has an interior or not.

To expand on the several nuances you consider :
Sauter dans un train can mean both jumping within a train and litterally jumping onto a train.

But a french speaker will probably understand from context that it was a figure of speech and that sauter dans meant you were in a hurry. The hurry may come from simply being late or, in your case, a last-minute decision.

Another option would be attraper le train ("catch the train").

[…] on a attrapé le train express vers Niigata […]

Qu’est-ce qui t’a pris d’attraper le train en marche ?

(The second one doesn’t sound as good though.)

Alternative to express how rushed you are: “Nous avons sauté dans le premier train pour Niigata”.

as a native french speaker i would use the verb “prendre”,

qu’est-ce qui t’a pris de prendre le train en marche

that sounds genuine, despite having the disgracious use of the same verb aside two times in a row.

Let’s not forget that when the slightly vulgar French speaker says “sauter dans un express” it is not at all a question of taking a train abnormally or as a hobo, that is without a ticket; this phrase connotes only the urgency that compels the person doing the action; a person whose action corresponds to the phrase almost always performs that action directly after a situation or happenning that motivates the hurry. That being said, a speaker of standard French would say rather “prendre immédiatement un express” or “prendre le premier express vers” according to case (and there are variants), but as Bob Dylan could once say, “Times, they’re a changing!”.