What does "he figures you" mean? I mean in this sentence: "He figures you get more overtime that way, being at a cheaper rate or whatnot."
"Figures" in this case means "to come to a conclusion". It essentially means that the person has used the premises that they have on hand to arrive at an understanding of the facts.
However, the phrase "figures you" is an actual expression that has a different meaning. If someone "figures you", it means they have decided you probably belong to some sort of group. For instance, "I figured you for a Yankees fan, not a Mets fan!", would mean the person thought that you were a fan of the New York Yankees, rather than the New York Mets.
In either case, "figure you" means they have come to some conclusion about you or your actions, but the case given is not the same as the expression "figure you".
"He figures (or believes) that you get more overtime...". This isn't really an idiom, just an elision.
As Zibbobz says, he figures you for a novice in some cases means 'he thinks of you as a novice'. This is an idiom, but not the one you are asking about.
Both these uses are informal at best, or regionalisms at worst; but they're worth learning.
You are parsing the sentence incorrectly. It is not that anyone is figuring you as the object of figuring.
What the person is figuring is that you get more overtime. In other words, it is a way of saying, "when he thinks about this situation, he thinks that you will get more overtime pay that way."