When the customer was paying through PayPal, my site's host down for a while. So, in the PayPal side I have receive the payment, while in the Amember side the customer hadn't finished the payment. What can I do with this situation? Can I manually add the payment history for that customer or let the PayPal send some information that make my Amember complete the rest work?
Yes you can manually add payment for this user. If there was temp. issue with your server, next payments should be handled automatically.