Finding variable importance and reduce dimension are different tasks. You can :
- Rank the variable according to the importance you suspect they have with respect to a target (sometimes referred as filtering). This could be: only retain the predictors correlated to the target, or rank them from a random forest importance scoring and keep the highest $n$% most important variables.
- Perform a "blind" dimension reduction, regardless of the target (random projections, PCA...)
As you don't have many examples, I suspect that filtering will lead to overfit. As your numeric predictors are correlated, scaling and dimension reduction with PCA will be a first step.
As for the categorical predictors, just remove the scarce levels if there are any. Per example, say you decomposed a variable age in levels (20-29,30-39,40-49,50-59,60+) and you discover that the category 60 years+ is represented only once. Then it will be better to merge it with the category 50-59 (calling it 50+) so that this category has more observations (and you dropped a category with only one observations).