Learn English – What does it mean to “have an air of importance”

A "run on a bank" happens when a large number of customers try to withdraw money in a short space of time. The mail is the mail-carrying horse-drawn coach the messenger is travelling on.

But Tellson's did not actually face a run: it was the bank messenger's imagination, driven by the sight and sound of the coach. Further on in the same paragraph:

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.

This web site says that "brought up by hand" means that he was bottle- or spoon-fed rather than nursed by his mother or by a wet-nurse.

By hand, brought up: Infants, in the absence of the mother, were either sent out to be fed by a wet-nurse (another lactating woman), or were spoon- or bottle-fed. Mrs. Joe's claim to neighborhood fame -- that she raised Pip "by hand" -- refers to the latter method.

This gave Mrs. Joe Gargery a "great reputation" because at the time, infant mortality was much larger among babies "brought up by hand". One can find usage of this original meaning of "brought up by hand", as well as discussion of the high resulting infant mortality, in Google books (the linked book is from 1846, a little more than a decade before Dickens wrote Great Expectations).