You seemingly don't have much access into this Mac mini, then.
You'll want to acquire screen sharing access with this hosted Mac (via macOS Screen Sharing or Apple Remote Desktop from a Mac, or from some VNC or RDP client if the hosted Mac is so equipped), and from there figure out why you can't gain access to the App Store to download Xcode, or from there figure out why you can't gain access to Safari to download Xcode from the developer portal.
Otherwise, you can install the command line tools from the command line. With ssh or telnet into the Mac or with access to Terminal.app, you can install the command line tools with:
xcode-select --install
If I understand the situation being described, this seems mostly a question about what macOS access the hosting provider is offering, and how. Check the docs from the hosting provider, as a start.