If you are new, I would not necessarily suggest you should change your delivery times to 5 days. Reason being, most of your competition is probably offering 2 days and under delivery. Now, this depends too, on what you are offering. Is it a service that does take more than 2 days to produce typically? If not, I would keep your delivery time at 2 days, as people are impatient, and only IF you can finish orders in that time frame, you would do better to keep the 2 day delivery.
As for this order, I agree that you should use the Resolution Center to ask for an extension and the reasoning (you have a drop down menu asking why you want to extend delivery) being you did not receive enough information from the buyer. Never deliver a half finished order to beat the clock. This can get you into trouble and possibly your account banned or suspended for breaking the TOS.