If your monitor is a genuine Commodore monitor, I remember that there were problems with some of them losing their picture after a while of use. If your A1200 is an NTSC model then you should be able to hook it up to a television the same way that you may have done with your C64 years ago. Alternatively, you could use the composite connectors to hook it up like a VCR. If you don't have stereo sound just hook it up to one of the sound jacks and the computer will auto sense mono audio for you.
Also, try holding down both mouse buttons when you start it. If the screenmode is incompatible with PAL or NTSC then holding down both mouse buttons will let you get to the early-startup menu. If it has a flickery screen it may be in PAL mode. If so press space to get it into NTSC mode.