I wasn't familiar with the passage but having read it I think it means that the sounds and sights of the nighttime mail carriage ride made the bank messenger fantasize a run (i.e. an excess of customer withdrawals) upon his bank. The wider context is that the shadows of the night are cast as various phantoms in the minds of the people on the carriage, and even the horse pulling it:
> While he trotted back with the message he was to deliver to the night watchman in his box at the door of Tellson’s Bank, by Temple Bar, who was to deliver it to greater authorities within, the shadows of the night took such shapes to him as arose out of the message, and took such shapes to the mare as arose out of HER private topics of uneasiness. They seemed to be numerous, for she shied at every shadow on the road.
> What time, the mail-coach lumbered, jolted, rattled, and bumped upon its tedious way, with its three fellow-inscrutables inside. To whom, likewise, the shadows of the night revealed themselves, in the forms their dozing eyes and wandering thoughts suggested.
> Tellson’s Bank had a run upon it in the mail. As the bank passenger—with an arm drawn through the leathern strap, which did what lay in it to keep him from pounding against the next passenger, and driving him into his corner, whenever the coach got a special jolt—nodded in his place, with half-shut eyes, the little coach-windows, and the coach-lamp dimly gleaming through them, and the bulky bundle of opposite passenger, became the bank, and did a great stroke of business. The rattle of the harness was the chink of money, and more drafts were honoured in five minutes than even Tellson’s, with all its foreign and home connection, ever paid in thrice the time. Then the strong-rooms underground, at Tellson’s, with such of their valuable stores and secrets as were known to the passenger (and it was not a little that he knew about them), opened before him, and he went in among them with the great keys and the feebly-burning candle, and found them safe, and strong, and sound, and still, just as he had last seen them.
I actually think it's a really neat passage, it invokes the surreal more than I would expect from 19th century prose. Also, this part just before is very vivid and quite funny:
> Except on the crown, which was raggedly bald, he had stiff, black hair, standing jaggedly all over it, and growing down hill almost to his broad, blunt nose. It was so like Smith’s work, so much more like the top of a strongly spiked wall than a head of hair, that the best of players at leap-frog might have declined him, as the most dangerous man in the world to go over.