It was really quick to do this today for me.

Here is the process that I used:

1. Ensure that whois privacy is disabled. You need to be able to receive emails as the admin contact before you can proceed.
2. Unlock the domain at GoDaddy (via your control panel).
3. Request the transfer authorisation code via email (via your GoDaddy control panel).
4. Log into your NameCheap account and set up a Domain Name transfer request for your domain name. It checks that it is possible. Then you can pay (using a discount code).
5. Watch your email box for the instructions for links that you need to click etc.
6. Look for the "pending transfers" page in GoDaddy to approve of the transfer immediately.
7. Click on a transfer update link in your NameCheap account to get things moving more quickly.

Very soon, you will be able to administer the transferred domain name.