The linked answer solution will work on Overleaf. Download a copy of apacite.dtx and apacite.ins into your Overleaf project folder from CTAN.
Then put the following as the first line of your document (you only need to do this once):
\immediate\write18{latex apacite.ins}
This will create a copy of apacite.bst in your project folder. Unfortunately it will be placed in the same directory as your log files and other files. If you select that file from the list in the "Other Logs and Files" menu you can download it to your computer and then re-upload it to your main project directory. I'm not sure if there is a way to do this directly.
I would rename it as advised in the linked answer. You can delete the .dtx and .ins files and the line above once you have successfully created the copy of apacite.bst.