One naive approach I can think of is to recursively eliminate features to exhaustively see which subsets of features has better metrics ($R^2$ score, etc). I think the key problem here is how to encode the categorical features and how to reverse back to original categorical features from those selected encodings after feature selection. But right now, the only encoding I know is one hot encoding which create so many dummy variables and I don't know any appropriate way of how to convert selected dummy variables to original categorical features (if this is the way to do it).
Solved – How to do feature selection on categorical features
categorical datafeature selection
Related Solutions
You could use boxplots to show the relationship between a categorical variable and a continuous output. In your case, you could make boxplots of the predicted price for the 5 districts.
Here is an example figure of what I mean
Of course how you create the box plots may depend on whether or not the districts are coded as 0's and 1's depending on what software you are using (I don't know what software you are using).
However, if the 0-1 coding is really what the problem is, I can't imagine it to be that hard to create another variable that is coded 1-5 based on the 0-1 coding.
One hot encoding would be a preliminary step toward dummy coding or effect coding or any other parameterization of a categorical variable. I don't know anything about scikit-learn (and questions about code are off topic here) but statistical programs such as SAS, R, SPSS, etc. do this encoding for you. It simply takes a single column of labels and turns it into k columns of 0's and 1's where there are k different labels.
You then have to choose what parameterization you want and which label you would like to use as your reference category. This has been discussed here before and will also be covered in any basic regression book.
Best Answer
This is common in NLP: manipulating very high dimensional feature vectors. Each dimension corresponds to a word (or a bigram) if you are considering a very simple case of text classification.
Feature selection, taking again the text classification task, can be done very naturally in Logistic Regression with $l_1$-norm. You control the strength of the parameter and the algorithm automatically prunes (sets their coefficients to 0) away features that don't contribute for the classification task.
In Bayesian setting, I believe feature selection is done using special priors like Laplacian. Unfortunately I am not very proficient in this field.
Having some custom rules to prune away features is nice and instructive. For a real problem, let a well-tested learning algorithm this do for you.