The batteries in the cartridges are not rechargeable, so I wouldn't think that leaving your Game Boy plugged in would bring them back to a full charge. My brother's Gold cartridge doesn't hold a save anymore, so I put it into a Game Boy Pocket that was connected to a power cord and left it turned on all last night. When I remove and insert the game and then turn it on, it still does not hold a game save.
As far as finding how much life is left in your game's battery, you could probably take a voltage reading across the battery's terminals and compare it with a graph of Discharge Characteristics from that type of battery's spec sheet. If you see that the voltage has dropped from 3V to around 2.5V (depending on the curve of the discharge graph), then the battery probably won't last that much longer. I Googled CR2025 and found one here: http://sanyo.wslogic.com/pdf/pdfs/CR2025.pdf