No, the Moon does not spin faster than the Earth. In fact, the Moon is tidally locked to the Earth, which means that it takes the same amount of time for the Moon to complete one rotation on its axis as it does to orbit around the Earth. As a result, the same side of the Moon always faces the Earth, and the Moon's rotation period is equal to its orbital period, which is approximately 27.3 days.
On the other hand, the Earth rotates on its axis with a period of about 24 hours, resulting in day and night cycles. So, the Earth's rotation is faster than the Moon's rotation.