Upload options are there only for the template:
/sitecore/templates/System/Media/Media folder
The options can be found under /sitecore/templates/Branches/System/Media node:
If you can see upload option as administrator and another user cannot see those options, there must be something with the access rights which does not allow that user to upload files.
Check Read access to /sitecore/templates/Branches/System/Media/Upload File and /sitecore/templates/Branches/System/Media/Upload File - Advanced items.