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I have a dataset that contains 20 predictor variables (both categorical and numeric) and one target variable (2 classes - Good and Bad). But, there are only 23 observations in the dataset. While I wait to receive significantly more observations, what tests / models can I perform on the available dataset to understand the variance between the good and bad cases, and to understand the variance within the cases classified as 'good'?

Ideally, for the data to make sense, I would want the variance within the good cases to be low, and the variance between the good and bad cases to be high.

Would multivariate analysis of variance (MANOVA) work in this case?

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I would start with logistic regression, and get a measure of importance for each of your 20 predictor variables. It's a little tough to understand what you mean by 'variance' in your two classes of good and bad. If you mean variance in your predictor variables that lead to your response, you can calculate the variance of a linear combination of predictor variables (in your logistic regression model) using an appropriate theorem from Wikipedia

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