I am not expert, but my own limited view is this. I said for thirty years I could see no interest in theorems like radon nikodym, having learned them very abstractly. then once while teaching honors calculus i asked myself what the fundamental theorem of calculus should say for integrals of just Riemann integrable, not necessarily continuous functions.
After originally stating the wrong answer to my class, I learned (with help from an analyst friend who showed me a Cantor function) that the indefinite integral is characterized by being a function which has a derivative equal to the original integrand almost everywhere (wherever that integrand is continuous) and being also not just continuous but Lipschitz continuous.
Then I realized at last the radon nikodym theorem is just the fundamental theorem of calculus for more general functions. We may not realize the analogy from calc 1, because by considering only continuous integrands there we miss out on the singular part. I.e. we forget to wonder why we are looking only at integrals of derivatives of C^1 (in particular Lipschitz) functions. So any application of FTC type is an application of RN, e.g. it lets you characterize constant functions (a.e?) by weak continuity and differentiability properties.
Similarly, Fubini is of course repeated integration, so reduces any integral computation inductively to one of lower dimension (volumes by slicing). E.g. to show some set has measure zero (as in Sard's theorem) you can do it inductively by showing most of the slices have lower dimensional measure zero. (See Guillemin and Pollack, appendix, or Milnor's differential topology book.)
One of my professors once suggested that virtually all problems in analysis are attacked by either dominated convergence or Fubini. So if you see an exam problem that dominated convergence won't do, try Fubini.
This answer assumes you retain an interest in the topic even after your exam, unless you are at Harvard, where exams are perhaps still in january.
Following up KCd's comment, you might also peruse old qualifying exams available on the websites at schools like Harvard and UGA. Harvard's also has a few lists of typical questions of this nature: e.g. if every intersection of a certain subset S of the plane, with a line of slope 1 is countable, what can you say about the Lebesgue measure of S? Gosh, Harvard even has, with login capability, online copies of all exams since 1977, and paper copies in libraries of exams since 1836!
Or consider a continuous weakly monotone increasing function f on the interval [0,1]. One knows that f is differentiable a.e. say with derivative g ≥ 0. If g is integrable and G(x) is the integral of g from 0 to x, then to what extent does G determine f or g, if either? When does G determine both f and g? Give an example if possible where G does not determine f, respectively g.