Save problem,maybe it's the battery??

Hi,I bought Riviera: The Promised Land catridge and I can save for
about 5 days. After that, suddently my save game becomes corrupt and I
try to play again and I can save but when I turn the GBA off,the save
data will be corrupt once again. So I opened the catridge,and I see
that the battery is rusted. So how can I change the battery? Cause
buying a new one will be a waste of money. Where can I buy the
battery? Is it a normal watch battery?? Or a normal mother-board(PC)
battery??? I bought this catridge from a shop. and I don't know if this
a fake catridge or not..I need help repairing this thing..thank you for
your help!