Can anyone tell me what happens in following situation:
Customer A's birthday is TODAY
We add them to list 1 TODAY
List 1 has an autoresponder which is set to send out an email the day / month before their birthday.
Am I right in thinking that Customer A won't get the auto response until the day / month before their birthday next year?
That's exactly right. The subscriber needs to have an active subscription at the time when the autoresponder email is to be sent, in order for the email to trigger.
Also make sure you use the "On the anniversary of a date" option for this, rather than "On the exact match of a date", if you do want it to trigger the next year.