How can I make Automator run service I made (shell script) and display the output in Terminal?
If you run a shell script by:


Opening the Terminal application
Telling the Terminal application to start your script


Then the script's output will, by default, be displayed inside the Terminal window (this can be changed by, for example, redirecting the output to a text file with the &gt; character). This is what you have been doing "before going to bed".

Shell scripts, however, can also be run without ever opening the Terminal application! For example, you could create a launchd plist to run a shell script whenever your computer finishes booting. In this case, the script would run, but nothing would be displayed on screen, because the Terminal application was never opened.

Similarly, Applescript's do shell script command executes shell code directly without ever opening the Terminal.

Instead of running the shell code directly, tell the Terminal application to run the shell code for you:

tell application "Terminal"
  do script "echo 'Hello World'"
end tell


Now, Applescript will open a Terminal window, and run your shell code inside of it.