How do I turn off notifications from Apple Arcade ?

“When you buy an eligible new iPhone, iPad, Apple TV, or Mac, new and qualified returning subscribers get three months free trial of Apple Arcade.”

You may get a message if you recently purchased a qualifying new Apple device. As this offer is time-limited, eligible users would be reminded before it expires (~1 week before).

The message is meant to entice user to redeem this offer, and start this service. Nothing happens if you do not redeem. You can ignore it or delete it, if you are not interested. If you get an on-device notification in Settings, then that should also have a Close [X] option.

See if your personal settings for emails from Apple are as you think they should be:

Manage emails that you receive from Apple - Apple Support