How to install SoapUI, open pre-prepared project, run tests and mock on ubuntu server?

There are different installers available for each OS platform.

In your case, a zip installer file can be used in headless ubuntu i.e., install in terminal by just unzipping it.

Under SOAPUI_HOME/bin, there is testrunner.sh utility is available. Use this utility to run/execute the pre-created soapui project in command line.