Setting up domain on VPS
Firstly, a nameserver is required however many domains you have pointing to one machine.
But you don't need to host it. You have 4 realistic options in total.
Most domain companies provide their own nameservers for you to use if you want. I'd be very surprised if GoDaddy didn't. Ask them.
Then there are third party DNS services like DynDNS or ZoneEdit. There are hundreds, if not thousands of services like this.
And then there's the possibility your VPS host has centralised DNS servers for their customers. I'm with Linode and they host all my DNS for me.
Failing that, you can always host it yourself on your server... If you only have one server, this is, frankly, a terrible idea. If the server dies (which they do from time to time) you lose very important DNS information (like MX email routing) as well as the website.
GoDaddy provides DNS hosting, so all you have to do is creating 2 A records, one for YOURDOMAINNAME.TLD and another one www.YOURDOMAINNAME.TLD, both pointing to your VPS's IP address