Hello anub_ms,
For difficulties with activating iMessage on your new iPhone 13, you'll want to take the steps below:
Check your device settings
1. Make sure that you’re connected to a cellular data or Wi-Fi network.
2. If you're using an iPhone, you need SMS messaging to activate your phone number with iMessage and FaceTime. Depending on your carrier, you might be charged for this SMS.
3. Go to Settings > General > Date & Time and make sure that your time zone is set correctly.

Turn off and restart iMessage and FaceTime
1. Go to Settings > Messages and turn off iMessage.
2. Go to Settings > FaceTime and turn off FaceTime.
3. Restart your device:
• iPhone
4. Turn iMessage and FaceTime back on.

Still not activated after a day?
It might take up to 24 hours to activate iMessage and FaceTime. If the issue continues after 24 hours, follow these steps:
1. Make sure that your device has the latest version of iOS or iPadOS.
2. If you're using an iPhone, contact your carrier to make sure that you can send and receive SMS messages.
3. Make sure you can send and receive SMS. If a prompt appears stating "Your carrier may charge for SMS messages used to activate FaceTime and iMessage", tap "OK" to allow possible SMS charges.
You can also find those steps in this resource: If you can’t turn on or sign in to iMessage or FaceTime - Apple Support
Should you still be unable to activate iMessage, you'll want to reach out to Apple Support: Contact - Official Apple Support They'd be in the best position to assist further.
Take care.