You say you're running Ubuntu Server, which usually means you aren't using any graphical frontend. But Copy&Pasting in Ubunu is managed by the Window Manager or Display Manager. So you can't really copy/paste there.
However, you could try enabling an SSH Server on the Ubuntu VM and then access it from the Windows console:
In VirtualBox's config: Select VM, then Settings → Network → Enable Adapter 1, attach to NAT.
In Ubuntu server:
- Install the OpenSSH server package:
sudo apt-get install openssh-server
- Configure the server according to the OpenSSH server guide
- Download and install PuTTY
- (Re-) Boot the VM
- Use PuTTY to connect to Ubuntu Server (you can get the IP from
ifconfig in Ubuntu's terminal). Use your Ubuntu username and password.
There you go. You should now have a Ubuntu Server terminal session in your Windows console. You can exit it using
logout or something in PuTTY.