Is it possible to share internet connection via remote desktop?

RDP is not capable of Internet sharing, so you can't share Internet using RDP with standard applications. There are many way to share Internet from a Windows box to another one. I suggest you to use VPN. You should do following steps:

  1. Run VPN Server on your server with unlimited internet access
  2. Connect to VPN server from your limited computer

  1. Install Bitvise SSH Server on your server.
  2. Use MyEnTunnel to make a SOCKS server on your computer. MyEnTunnel uses SSH to pass SOCKS request to the SSH server.
  3. Use Proxifier to redirect any applications traffic to the SOCKS sever created by MyEnTunnel. This step is not required if the update program accept SOCKS.

You could try upgrading your PC using DVD or USB media instead of an online install. If you can't obtain the DVD elsewhere you could still download the media image using the server and then transfer it to a local PC that has a DVD burner (possibly using Remote Desktop's drive sharing but using a shared folder on the network would be better).