Yes you can pipe your Internet connection over the virtual machine's. However, the speed of your Internet will be the slower of the VM's Internet and your Internet, minus a little overhead. So in that respect, no you can't take advantage of the VM's Internet speed while at home.
That said, if you are able to do the activities you want remotely on the VM instead of at home, it will be at the speed of the VM's Internet. So for example, if you installed Windows Pro or Ultimate on the VM and connected to it via remote desktop, you could use it over a 1 Mbps connection from home. Stuff you downloaded to the VM from the Internet would transfer up to its full 1 Gbps connection. Basically the VM is connected to the Internet at 1 Gbps, while its desktop display is connected to you at 1 Mbps. As long as nothing is going on with the display which exceeds 1 Mbps (no games or video), it'll be almost like using a local system with a 1 Gbps Internet connection. Except your data is stored on the VM instead of on your local system.