The custom metadata is present in the broker only to allow filtering and not as pure "model" data; as such the APIs do not provide you with a method to directly lookup this data via a normal API call (you can create filters based on custom meta to get components that have certain custom meta field values set).
THe "model" data that is exposed from the broker is the content of the relevant dynamic component presentation. The solution would therefore be to have the relevant metadata available in the rendered content of the component presentation (you can even have a dedicated CT for exposing this metadata so that your "normal" published content doesn't get polluted with this metadata).
Update: if the metadata values are metadata on a keyword and the taxonomy is published this post should explain how to retrieve metadata from published keywords. There should be no need to publish as DCPs instead.
In your case below code might work:
TaxonomyFactory taxonomyFactory = new TaxonomyFactory();
Keyword keyword = taxonomyFactory.GetTaxonomyKeyword("tcm:38-979-1024");
CustomMeta customMeta = keyword.KeywordMeta;
var theValue = customMeta.GetValue("case_icon");
