Hi yevyapan,
Welcome to Apple Support Communities. I understand that you're having trouble when you try to FaceTime a specific contact, but when they contact you there isn't an issue. I know how important it is to use FaceTime; I'd like to help. Since you mention this behavior only happens when you try to contact them, I'd recommend that both you and the contact check to ensure your information isn't blocked:
To see the phone numbers and contacts that you've blocked from Phone, FaceTime, or Messages:
Go to Settings > Phone > Call Blocking & Identification.
Go to Settings > FaceTime > Blocked.
Go to Settings > Messages > Blocked.
From these screens, you can add or unblock contacts or unblock phone numbers.
To add a contact from Phone, go to Settings > Phone > Call Blocking & Identification > Block Contact. Then tap the contact that you want to block. To add a contact from Messages or FaceTime, go to Settings > Messages or Settings > FaceTime, scroll down and tap Blocked, tap Add New, then select the contact that you want to block. To unblock a contact or phone number, swipe left over the number, then tap Unblock.
Block phone numbers and contacts on your iPhone, iPad, or iPod touch
If you continue to have trouble, I'd also suggest that you have the contact remove you from their Contacts app and then add your information back and test again.
Let me know what you find.
Best Regards.