Please please help:my iPhone 3GS 32gb on iOS 6.0.1keeps turning off unexpectedly even when charge is 100% and cannot power on unless connected to power...please help here

Since a 32GB iPhone 3GS was made only when this phone was Apples flagship phone, it was made sometime before June 2010. Given that age minimum and the symptoms you are experiencing, I would guess that the issue is that the battery needs replacement. You can have Apple do this, have someone else do it, or do it yourself, in order of decreasing price.

Take it in to an Apple Store if you can. The genius bar can very quickly test if it is just the battery showing its age or not. If it is the battery, you can get it replaced through Apple

http://www.apple.com/batteries/replacements.html

Depending on how it has been used though, it may not be the battery. My old iPhone 32GB 3Gs is still good for about 6-7 hours of continuous use (as an iPod touch these days) and it was bought new in 2009 shortly after the 3Gs became available at AT&T stores. So yours might be experiencing some hardware failure not the actual battery itself.