A 1-d problem, so ONE unknown? Don't use fmincon at all!
And if you wish to supply the gradient, then you cannot supply only an approximate gradient. That will just cause problems. Fmincon does not have an option where you can give it hints.
You MIGHT consider fminbnd, a tool designed to solve 1-d problems.
And of course, since you have a stochastic problem, in the sense that it will not even be differnetiable near the solution, again, fmincon is a terrible choice of tool. fmincon will assume differentiability. It sill assume your objective will return a consistent value, so if you evaluate the function a second time at the same point, it should return the same value. So even fminbnd will have issues due to a stochastic objective.
I would probably just write my own code, where I use the last few objective evauations to form a local polynomial approximation to the problem. At the same time, you can use that modeling to determine an approximation for the level of noise to be expected. And that will tell the algorithm when to quit, that further searching will not yield a better approximation to the solution.
Finally, could you use such a tool to include prior information on the approximate gradient as a hint in the search? Well, yes, you could. Now the problem will start to look like a Kalman filter. Estimate the coefficients of a local quadratic polynomial, given some prior information about the derivative of said polynomial. Then find the minimum of the polynomial as your next iteration. Since you also have information about the uncertainty in your coefficients from the kalman filter, that will tell you how much you can trust that next step, and tell you when to stop looking. This is code you would write yourself of course.