Apple Arcade isn't appearing in my App Store app on my Mac

You need macOS 10.15 for Arcade to show up.

here is an excerpt from this link - Join Apple Arcade - Apple Support

Before you join

Make sure that you have an iPhone or iPod touch with iOS 13, an iPad with iPadOS 13, a Mac with macOS 10.15, or an Apple TV with tvOS 13. Then follow the steps below to join Apple Arcade.

Open the App Store

On your iPhone, iPad, iPod touch, Mac, or Apple TV, go to the Arcade tab of the App Store and look for the subscription offer. If you don't see it, try to download an Apple Arcade game, and you'll be prompted to subscribe.

Start a trial

Tap or click the trial subscription offer. (One trial per Apple ID.)

If you're using Family Sharing and someone in your family group subscribes, you'll see a message that you have access to the subscription if you try to subscribe.

Sign in with your Apple ID and password

You might be asked to enter your Apple ID and password. Make sure that you sign in with the Apple ID that you use for App Store purchases.

If you don't have an Apple ID, follow the prompts to create one. If you're not sure if you have an Apple ID, we can help you find out.

If you're asked for a verification code, you have two-factor authentication on your Apple ID. Learn about two-factor authentication and what to do if you still can't sign in.

Confirm your details

Make sure that you have a valid payment method set up for your Apple ID, and agree to the terms and conditions.