The Power BI API returns admin info on the state of a dataset, but not the data structure.
If the dataset is compatible and the XMLA endpoint for the workspace is open, you can connect to that and use it like you would an Analysis Services database, including getting the measures and columns. You can use any tool or custom code that implements the MSOLAP or AMODB libraries. DAX Studio is just one of these options. The url of a workspace will be powerbi://api.powerbi.com/v1.0/[tenant name]/[workspace name]
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