Feature Importance without Random Forest Feature Importances
There are many ways to try to estimate feature importance. Personally I think the random forest measures get overused simply due to the fact that they have “importance” in their name and many people have heard of them. However, what people don’t realize is that those features that the random forest deems important are important for the random forest. They are good at predicting your feature of interest in a random forest setting. Taking this and blindly applying it to non random forest problems is dangerous. What might be predictive for a random forest may not be for some other algorithm. Also these random forest importance measures are not without their faults, for example they are biased towards variables with wide ranges.
There are many other methods available for variable importance such as information gain and relief. I suggest you read this paper by Robnik-Sikonja
https://link.springer.com/content/pdf/10.1007%2F978-3-540-39857-8_30.pdf
It covers many different methods.