Yeah, check and double check all caps and polarity, and soldering. If all else fails, redo the cap job. I did a Game Gear a few months ago that wouldn't power on after I did a cap job, it turned out the new caps in the power board were bad.
*edit* Now that I think about it, my Game Gear did EXACTLY what your GT is doing. Would power up for one second, then die. I put the old power caps in and it turned right on.