The sound IS on your CloneDVD backup. Sin City has a DTS soundtrack and a Dolby track. If your home entertainment system doesn't decode or is not set to decode (in player or amp settings) DTS then just use the the Dolby track which you said is also included on your backup. Change sound tracks via the DVD's setup menu or do it with the "audio" button on the players remote control ... press the button several times to cycle through all the tracks till you get the track that plays.
You must "Preserve Menus" in CloneDVD. On this movie, the sound and subtitle tracks are authored to matrix throught the menu structure and if you have not kept the menus, your DVD player is not receiving default audio or subtitle information from your backup and so it chooses it's own default ie DTS and CC.
BTW, since you already own CloneDVD and AnyDVD, you don't need to use the freewares, Clone & Any handle this movie just fine. It's just your methodology that's incorrect. Also by using multiple programs together they may be cancelling out each others features considering the subtlty of the issue of default audio and CC subtitle playback. But I understand if you are just fond of Shrink ... many are and it's a long shot that Shrink and CloneDVD are conflicting.