As it's heating up do you hear any gurgling coming from the cooling unit? Do you see any areas on the cooling unit that appear to have a yellow stain? Either one would indicate cooling unit failure.
Turning it upside down is fruitless if the unit has failed. It's a lot cheaper to replace the cooling unit than the entire fridge, and a new cooling unit usually carries a lot longer warranty than a new fridge. But you may not be able to buy a cooling unit for a fridge that old.