S.S.
Why will your neighbor be the one to bring her? I was worried about this with my oldest - who was PAINFULLY introverted... I made arrangements to go with him the first week. Paid and everything, went to class - played for 10 minutes, watched him play for 20 minutes, waited until he was REALLY engrossed, and then asked if he was ready to go. He said no, so I told him I had to go clean up and I'd pick him up later. We've never looked back.
Had he said yes, I would have taken him home and it would be nap time.