I don't think the Sh. eq. requires two years of calculus to understand, but it might require that to solve it in novel situations. Indeed, for that it's probably even more important to have a deep understanding of algebra (in the way mathematicians use the term) than calculus! But even without that you can begin to understand it if you understand its structure, without knowing how to solve it in general situations. The full Sh. eq. is a partial differential equation because it depends on both x and t, but the usual trick to solving it reduces it quickly to an ordinary differential equation that depends only on x, which knocks a year off the calculus right there!
The trick is to look for "energy eigenfunctions", which have "stationary" behavior (which really means their magnitude stays fixed and their phase advances linearly in time, like the hands of a clock). You can then build up any general solution as a linear combination of eigenfunctions that are each just going around like clocks, but they have complicated spatial behavior that can do whatever you need when you superimpose them to fit the initial conditions you are given. The spatial behavior is what you need calculus for-- you replace the full Sh. eq. with the "time independent" Sh. eq. by asserting you are just solving for the spatial dependence of each "energy eigenfunction". That means you can replace d/dt by -omega (that's the clock business) and the rest of the Sh. eq. is now an ordinary differential equation in x, for the eigenfunction that corresponds to that omega. It might not work for just any omega, but when it works only for discrete omega, algebra theorems tell you that you have a discrete spectrum of eigenstates that you can expand any general solution in terms of. So finding those eigenfunctions (functions of x) is the tricky part, but you can just look at the tricks used, you don't really need to be able to solve it yourself to understand either what the equation is doing, or what is the nature of the eigenfunction. That is already a pretty useful form of understanding of the Sh. eq.