It looks like your customer asked you to estimate delivery date but not an effort in hours. In other words, in case of fixed price and specified job, a customer may negotiate only delivery date. A developer (you, in this case) needs to estimate an effort and decide how to put this effort into timeframe to secure comfortable delivery date. It is also worth thinking whether the estimated effort is actually payed off by the fixed price. Plus, if your customer wants delivery earlier, you may consider to ask for bigger price and hire second developer to help you and pay him. The book "Software Estimation: Demystifying the Black Art" explains this 'science' is details.
If your customer is really interested in estimation of effort, I would ask him why he needs it...