Hi & welcome to BBF
would you by any chance be using msn on your pc? hotmail will not work when msn is in use on your computer.
If that is not the case, try deleting then readding your email address on o2's Blackberry Internet service webpage.
Was your battery fully charged when you got it? I tend to Leave my phone on when charging, the red zigzag disappears from the battery level indicatorwhen fully charged.
Do you have any applications running in the background?, make sure you close all apps by hitting the menu key and selecting close.
The GPRS is just the signal and will not affect battery life or use your allowance.
Hope that helps