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It's not possible since the permission Approve Items comes with a dependency to the following permissions: Edit Items View Items View Pages Open Source: User permissions and permission levels in SharePoint 2013
There is no "Publish" you just save the document and it will default to the source Library. If they edit it in the browser it will save back to the Library as well. The user has to have sufficient permissions to perform the task, in this case they need Edit rights.
You need to install Microsoft Office 2010: Primary Interop Assemblies on the machine you're deploying; here is a link to Office 2010 in-case you're using Office 2010-> http://www.microsoft.com/en-us/download/details.aspx?id=3508
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