Whack-a-Mole was a minor side mission in Ōkami, taking place in Taka Pass. After entering Taka Pass, Amaterasu met a tea master who complained the Molsters had stolen his beloved teacup, prompting Amaterasu to try and get it back.
Walkthrough
The moles can be found nearby in the pass once the cursed zones have been lifted by restoring the Guardian Sapling, and will promptly start a cutscene in which they accuse you of trespassing. Afterwards the fight start proper, which is a variant on the classic Whac-A-Mole game. There are six mole holes, five moles, one of which is Moley, their leader that wears a purple cap and blue bag on his back. All of them will dig underground and travel to one of the six mole holes, at which point you are to find Moley and tackle him.
Celestial Brush techniques will be ineffective (with the exception of Cherry Bombs, though the timing will be tricky), and hitting any of the Lil' Moleys also does nothing as they resurface each time the rest goes underground. In total Moley must be hit three times, which becomes progressively more difficult after each hit. Make sure to watch the ground carefully when the moles are traveling around to predict where he will pop up. Once finished he hands you the stolen teacup, which you must carry to the tea master. In return he gives you the Golden Mushroom, which will only be of use some time later in the side mission "Serve up the ultimate dish".