I would say to get it running try and use NAT first.
Check that Virtualbox is NATing/Bridging you wireless card and not the wired adaptor.
Does your VM have any network devices?
Does the network device have an IP address?
If you do try using the ping cmd from a terminal shell to check you can reach your router. e.g. ping router_ip_address
I have not used VirtualBox, I use VMware Player, so cant help too much with the specifics.
Adrian