The Ideal time frame should be around 5 days (working days).
How can I overcome this?
It's up to you to figure out how to overcome a requirement. Perhaps you could talk the customer out of that for version 1. Then, just don't plan a version 2.
Or should I try to put the shield to sleep?
That isn't possible. Cutting power to it might be. It depends on how you feel about bending pins.
You need to get out your handy-dandy calculator. How much current are you drawing? For how long? That tells you how many times you can do something before the battery goes dead.
Are you planning on modifying the GSM shield? If not, then the shield draw (a lot) of power all the time, and more when it is actually sending.
The PIR sensor will draw power all the time.
A switch that triggers an interrupt, activated when the mail door/slot is opened could wake the Arduino from sleep mode, greatly extending battery life.
This project seems like a waste of money. Buying a SIM card just to know when the mail is delivered?
How far is the mailbox from you? There may be cheaper/lower power consuming ways to know when the mail arrives.