I just purchased and set up a Linode VPS. Normally I purchase a new domain name and set it up with my hosting but on Linode didn't ask for that. What I have to do to publish my web application to Linode?
Linode is not a managed service. This means you have to set everything up from a-z.
They have a good range of stack scripts to setup your server. Then you'll need to add your domain using the DNS manager of the linode manager page. Then you'll probably need to manually configure your server's config files to add your domain.
It's a bit of a process if you're new to it all, but that's actually the benefit of an un-managed VPS, having full control of the system.
If you're new to this, and didn't realise what you were getting into, it might be best to lodge a support request for a refund. And next time look for a Managed VPS. Managed means someone will set it all up for you.
Hope this helps.