It's called What About Me by Moving Pictures, an Australian 80's band.
Bumper music is basically the music radio stations play transitioning between programs or commercials.
I was 12 years old at the time and remember it well. Memories. We need to bring back the power ballads!
There is also a modern (country) version also by a fellow Australian Shannon Noll. Shannon was the runner up of the 2003 Australian Idol. Not as good as the original but still good. http://www.youtube.com/watch?v=kqyIwZpr5y0