Meaning of 'settle up with'

In this context "You can settle up with Jane or John." means "Jane or John will pay your bill". To "settle up with" is to come to a final settlement of a financial obligation, agreeing on the amount do an paying it or arranging payment of it. Most often it simply means "pay the bill" but it carries an implication of "render an account and get it paid" that is, it includes delivering a bill, often a more or less detailed bill, coming to an agreement on what is owned (if there is any dispute), and making payment.

No joke seems to be intended here.

The phrase is also used metaphorically, to mean getting satisfaction or revenge. Fort example 'John insulted me, but I am going to settle up with him before long." The implication here is that the speaker "owes" John something for the insult, and with deliver what is owed, probably some sort of revenge or retaliation. This metaphorical use is something of an idiom, but is not the use intended in the quoted text.