I have an ubuntu instance with mono installed and would like to install a simple api project that is on another computer.
I do not have the "Web Deployment Settings" options referenced in this article. http://www.monodevelop.com/documentation/creating-aspnet-projects/
How do I move my project to the server and start it up?
If you want to do this properly you might want to look into deployment tools such as Capistrano or Chef.
First, you'll want to create your Mono app as a console application. Linux can just run your executable, there isn't an easy web host environment like IIS for Mono apps (yet). This of course means your app could exit on an exception, so ensure you have adequate defensive coding to avoid that happening.
Basically, to move the files you can use SCP, just scp -R <directory> <username>@<server>:/<path_to_copy_to>. Once you've got the files there you then need to start the app, and I'm guessing you want to run it as a service, not just on-demand. In order to do this you'll need to create an Upstart script. You'll want to create a job for starting, stopping and maybe restarting your app.
If you add it to the right runlevels it will automatically start on boot and stop on shutdown.