Introduction
According to Microsoft, A content type is a reusable collection of metadata (columns), workflow, behavior, and other settings for a category of items or documents in a Microsoft SharePoint Foundation 2010 list or document library. Content types enable you to manage the settings for a category of information in a centralized, reusable way.

Here I'm assuming that you have content type hub already setup at your end. For more information on how to setup content type hub please find this link ( http://sharepoint-community.net/profiles/blogs/understanding-content-type-hub-cth-in-sharepoint-2013 ).
Information :
Please find below is the powershell script which will help you publish the content type.
  1. $CTHubURL = "{<Content Type Hub Setup URL>E.g. http://mySharePointSite :1111/}"
  2. #Get Content Type site and web objects
  3. $ctHubSite = Get-SPSite $CTHubURL
  4. $ctHubWeb = $ctHubSite.RootWeb
  6. #Check the site is a content type hub
  7. if ([Microsoft.SharePoint.Taxonomy.ContentTypeSync.ContentTypePublisher]::IsContentTypeSharingEnabled($ctHubSite))
  8. {
  9. #Set up ContentTypePublisher object to allow publishing through the Content Type Hub site
  10. $spCTPublish = New-Object Microsoft.SharePoint.Taxonomy.ContentTypeSync.ContentTypePublisher ($ctHubSite)
  12. #Step through each content type in the content type hub
  13. $ctHubWeb.ContentTypes | Sort-Object Name | ForEach-Object {
  15. #Has the content type been published?
  16. if ($spCTPublish.IsPublished($_))
  17. {
  18. #Republish content type
  19. $spCTPublish.Publish($_)
  20. write-host "Content type" $_.Name "has been republished" -foregroundcolor Green
  21. }
  22. else
  23. {
  24. write-host "Content type" $_.Name "is not a published content type"
  25. }
  26. }
  27. }
  28. else
  29. {
  30. write-host $CTHubURL "is not a content type hub site."
  31. }
  32. #Dispose of site and web objects
  33. $ctHubWeb.Dispose()
  34. $ctHubSite.Dispose()