I've found a way to read the value using the TFS 2013 API, inside the ISubscriber.ProcessEvent method:
var workItemId = 12345;
var extService = new WorkItemTypeExtensionService();
var workItemService = new WorkItemService();
var wit = workItemService.GetWorkItem(requestContext, workItemId);
foreach (var wef in extService.GetExtensions(requestContext, wit.WorkItemTypeExtensionIds))
{
foreach (var field in wef.Fields)
{
if (field.LocalName == "Kanban Column" || field.LocalName == "Backlog items Column")
{
// Access the new column name
var columnName = wit.LatestData[field.Field.FieldId];
}
}
}