One way to think about the process of building a predictive model (such as a neural network) is that you have a 'budget' of information to spend, much like a certain amount of money for a monthly household budget. With only 87 observations in your training set (and only 36 more in your test set), you have a very skimpy budget. In addition, there is much less information in a binary indicator (i.e., your predicted variable is positive vs. negative) than there is in a continuous variable. In truth, you may only have enough information to reliably estimate the proportion positive.
Neural networks have many advantages, but they require very large numbers of parameters to be estimated. When you have a hidden layer (or more than one hidden layer), and multiple input variables, the number of parameters (link weights) that need to be accurately estimated explodes. But every parameter to be estimated consumes some of your informational budget. You are essentially guaranteed to overfit this model (note that this has nothing to do with the computational feasibility of the algorithm). Unfortunately, I don't think cross-validation will get you out of these problems.
If you are committed to building a predictive model using your continuous variables, I would try a logistic regression model instead of a NN. It will use fewer parameters. I would fit the model with probably only one variable, or at most a couple, and use the test set to see if the additional variables (beyond the intercept only) create instability and reduce your out of sample accuracy.
Regarding the X variables themselves, I would use a method that is blind to the outcome. Specifically, I would try principal components analysis (PCA) and extract just the first one or two PCs. I honestly think this is going to be the best you are going to be able to do.