Easy way to execute shell script
Appifying with Platypus:

Download Platypus and create an app from the shell script.

Example:



The settings above create an app with a progress bar (including details) and it can be quitted - the rsync process has to be canceled (killed) manually though. 



The app itself can be found with Spotlight.



Simple AppleScript solution:

In Mavericks (and probably later) you have to allow the AppleScript app to control the computer in "System Preferences -> Security &amp; Privacy -> Privacy -> Accessibility"

Open ScriptEditor, enter the code snippet below and export it as an application with a unique name:

tell application "Terminal"
    delay 2
    set newWindow to do script "caffeinate -s rsync -hva --progress --stats --delete /path/to/folder /path/to/folder2"
    activate
end tell


If you need root permissions prepend sudo -> "sudo caffeinate -s ...."

You can refine the AppleScript app by checking if Terminal already has open windows and create a new one to run the shell script:

set myScript to "caffeinate -s rsync -hva --progress --stats --delete /path/to/folder /path/to/folder2"
tell application "System Events"
    if exists (window 1 of process "Terminal") then
        tell application "Terminal"
            set newWindow to do script myScript
            activate
        end tell
    else
        tell application "Terminal"
            delay 2
            set Window1 to do script myScript
            activate
        end tell
    end if
end tell


The AppleScript app can be found with Spotlight.



In Mountain Lion you have to enable "System Preferences -> Accessibility -> Enable access for assistive device" and change it a little bit to get the AppleScript app solutions running properly:

set myScript to "caffeinate -s rsync -hva --progress --stats --delete /path/to/folder /path/to/folder2"
tell application "System Events"
    if exists (window 1 of process "Terminal") then
        tell application "Terminal"
            reopen
            activate
            set newWindow to do script myScript
        end tell
    else
        tell application "Terminal"
            reopen
            activate
            do script myScript in window 1
        end tell
    end if
end tell




AppleScript with "external" shell script (backup.sh):

This is the final approach the OP is using. Taken from here. In Mavericks (and probably later) you have to allow the AppleScript app to control the computer in "System Preferences -> Security &amp; Privacy -> Privacy -> Accessibility". One can load the full .sh (placed inside the app in Contents/Resources/backup.sh) with:

set bashFile to path to resource "backup.sh"
tell application "Terminal"
    delay 1
    activate
    set newWindow to do script "bash " &amp; quoted form of (POSIX path of bashFile)
end tell


You can refine the AppleScript app by checking if Terminal already has open windows and create a new one to run the shell script:

set bashFile to path to resource "backup.sh"
tell application "System Events"
    if exists (window 1 of process "Terminal") then
        tell application "Terminal"
            delay 1
            set newWindow to do script "bash " &amp; quoted form of (POSIX path of bashFile)
            activate
        end tell
    else
        tell application "Terminal"
            delay 2
            set Window1 to do script "bash " &amp; quoted form of (POSIX path of bashFile)
            activate
        end tell
    end if
end tell