Hi there, I recently bought a Classic Gameboy from a "flea-market" and I cleaned it and put new batteries in and it works...

Did you check that nothing is blocking contacts inside the game-cartridge or in the Gameboy Micro? You can try a little bit of contact spray or to cleanse the contacts first with something like "Kontakt LR" or a similar product. Kontakt LR (www.crcind.com) has a brush included to be able to clean the contacts.