Doing a cap replacement on a Game Gear is horrid. I have six of them at home waiting for me to sell them.
The first problem is getting the old ones out. They're can caps, but the can is encased in a plastic cover, and the cap is folded over the solder points. Removing the caps makes your house stink like burning fish... and hopefully you don't lose the pads to either removing the cap, or bad electrolyte eating the pad away.
Then the next part involves putting in new caps. If you use regular can caps, you have to make sure they're not too wide or else you can't get the unit back together when you're done.
That doesn't even include the caps on the sound board or the power board. Those are even worse to get at. Don't get me started on that sound board.
Having said all that, how busted is the screen? Is it just dark? Maybe the backlight bulb is on it's way out. You might be interested in removing the backlight and replacing it with a couple white LEDs and a drinking straw.

Here is the schematic for the backlight section. Easiest way I can think of is to remove everything in the top right corner. Positive lead goes on the positive side of L2 (or R55/R57), and the negative lead goes on the negative side of C68. Use an appropriate resistor for the LEDs, and the contrast dial should still work afterwards.