I always start by cleaning the contact patches on the board and the rubber domes below the buttons. They're prone to getting corroded and/or packed full of crud if the Gameboy (or controller, or whatever) has been stored carelessly for a long time.
I've fixed several with that issue.
Is there any visible corrosion on the board? Any damage?
If not, then the contact points probably need a good cleaning with rubbing alcohol.
You can also try cleaning the silicone pads with rubbing alcohol, but you might need new pads.
Retrospekt does repairs I believe. They also have gameboys for sale.
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