You will need to leave the PC on in order to use Redirector.
If you are a remote user (outside sales for example), then you are probably using a VPN to connect to the corporate servers, and the VPN probably has a time-out limit that will automatically log you out and disconnect your Outlook then Redirector will have nothing to redirect.
Also, I am going through a similar situation (Regional Managers and above are given Blackberries) and found out that corporate policy does not allow personal equipment to connect to the BES.
So I am using my personal e-mail accounts on the Blackberry, and will continue to schlep my laptop with aircard when I am on the road for the company.
I can't wait for my company to issue us stumps a Blackberry, then I will have TWO!