You can do a SOAP Post call leveraging DataExtensionField object.
The SOAPAction should be "Retrieve"
The payload should be something similar to:
<?xml version="1.0" encoding="UTF-8"?>
<s:Envelope xmlns:s="http://www.w3.org/2003/05/soap-envelope" xmlns:a="http://schemas.xmlsoap.org/ws/2004/08/addressing" xmlns:u="http://docs.oasis-open.org/wss/2004/01/oasis-200401-wss-wssecurity-utility-1.0.xsd">
<s:Header>
<fueloauth xmlns="http://exacttarget.com">{{access_token}}</fueloauth>
</s:Header>
<s:Body xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns:xsd="http://www.w3.org/2001/XMLSchema">
<RetrieveRequestMsg xmlns="http://exacttarget.com/wsdl/partnerAPI">
<RetrieveRequest>
<ObjectType>DataExtensionField</ObjectType>
<Properties>FieldType</Properties>
<Properties>Name</Properties>
<Properties>IsRequired</Properties>
<Properties>Scale</Properties>
<Filter xsi:type="SimpleFilterPart">
<Property>DataExtension.CustomerKey</Property>
<SimpleOperator>equals</SimpleOperator>
<Value>DE_CUSTOMER_KEY</Value>
</Filter>
</RetrieveRequest>
</RetrieveRequestMsg>
</s:Body>
</s:Envelope>
You can get use more properties on your payload to get more information about your fields, like:
<Properties>DefaultValue</Properties>
<Properties>MaxLength</Properties>
<Properties>CreatedDate</Properties>
More properties to be found in the link to the DataExtensionField documentation provided above.
PS: Don't forget to add data_extensions_read
scope to your authentication payload, and to your package on Marketing Cloud as well.
For a step by step guide, checkout Gregory's latest article: Introduction to making API calls to SFMC from POSTman