I would argue that you probably need the gist of at least four or five post-calculus mathematical topics usually taught in one semester college courses, although not always lumped together in the packages.
* Differential equations
* Linear algebra
* Applied analysis with a hodgepodge of tools such as tensors, spinsors, integration with Dirac Delta functions, and the calculus of variations and path integral evaluation
* Complex analysis
* Abstract algebra and
* Topology
(Most string theorists would have a taken an advanced real analysis course as well, but while necessary for rigorous work, to get the gist of the ideas this can probably be skipped almost entirely, as can higher level mathematical probability and statistics that is crucial for experimental work and careful calculation but can be dispensed with if you just want the gist of it.)
I would also argue that you need the gist of at least two or three upper division/early graduate school level physics courses introducing the subjects of:
* General relativity
* Quantum mechanics (probably a two semester set of courses to get sufficiency master of not just QED and Dirac's Equation, but also QCD and electroweak unification).
* An advanced mechanics class, even if its classical mechanics (e.g. three body problems), to develop a mastery of concepts like Hamiltonians and Lagrangians.
So, you need the gist of six to eight three or four semester credit college courses beyond your basic calculus and physics, which would be something on the order of 25 weeks of full time study if you took the courses themselves and passed them.
But, of course, you can get the gist of those subjects well enough to make some sense of published works and understand what string theorists are talking about without doing research and using the equations to make more than back of napkin calculations on your own with far less work. You could probably pick up all the physics you need to get the gist of it in the equivalent of the work that goes into a single three credit mid-level undergraduate survey course (perhaps 45 hours of real time). You could probably get the gist of all the math you need other than abstract algebra and some of the finer topological points in a similar amount of time or maybe even a hair less (call it 35 hours). But, while you don't need all of either a rigorous single semester course in either abstract algebra or topology to get the gist of string theory, you do need a quite a bit higher level of mastery of each of them to get even the gist of those concepts which are not very natural ones to someone with your level of mathematical and physics background, in part because they involve a lot of non-intuitive terminology. You probably need 20-30 hours of work to get enough of the gist of these topics to be useful. After that it would probably take an effort to read a few dozen academic articles and conference presentations in the area pretty carefully, at twenty minutes or so each to pick up on string theory specific conventions used in the field. So maybe that would take 15 hours.
So, overall, it probably takes 115-125 hours of study to get even the basic gist of string theory well enough to follow the main points made in a published string theory paper or conference presentation about string theory taking a position that you will trust the presenter or author to be correct in his or her technical mathematical manipulations but educating yourself enough so that you could understand and articulate the main conclusions of the presentation, the basic program of the reasoning used to get that result, and to have some sense of why someone might ever care about the questions asked in the first place.
Of course, with lower standards for the "gist" of that you want to achieve (e.g. simply to be able to say what string theory is as a program and its main subdivisions and most well know premises) you could probably master that in less than a week (i.e. less than forty hours) of browsing wikipedia articles and referring back to old textbooks and maybe reading a short quantum mechanic's treatment like Feynman's QED and skimming one or two popular book length treatments of string theory intended for educated laymen.
(I have improved my knowledge in several other areas too, including linear algebra, abstract algebra, topology, measure and integration theory, functional analysis, differential geometry, Lie groups, quantum logic and general relativity).