Backpropagation on a matrix of functions
The problem isn't well-defined as you've stated it.  Backpropagation involves updating the weights, i.e., finding weights that cause the neural network to behave well.  If you have no weights, then there's nothing to update.

For these reasons, the standard approach to building neural networks involves both weights and univariate functions.  In the standard perceptron architecture, there are weights $w_{ij}$ along each edge (from neuron $i$ to neuron $j$), and a univariate function $f_j$ applied to the output of each neuron (neuron $j$).  So standard artificial neural networks already incorporate the univariate function.  The univariate function $f_j$ is called an "activation function".

Standard backpropagation does work with arbitrary activation functions.  You don't need to modify or generalize it: it already deals with them.  In particular, backprop can be used with an arbitrary activation function, as long as you know its derivative.  So, I suggest you take another look at how standard backprop works.

Backprop is basically just (a) computation of the partial derivatives (the partial derivative of some objective function, with respect to the weights), computed using the chain rule, plus (b) gradient descent.  Introducing arbitrary univariate functions affects only part (a), and the chain rule from calculus tells you how to calculate the partial derivatives.