I'm missing some information:
- Is your phone rooted?
- Do you have USB debugging enabled?
- Are your contacts synced to your Google account?
Since it isn't clear from the question - have you checked that your contacts aren't already backed up to your Google account? If you had your contacts synced - then they should all be backed up to the cloud / your connected Google account.
If that's not the case, and all you need is to be able to see your screen and control your phone (via the app of your choice) - you can try this kind of solution - an an MHL-enabled USB port.
Also, maybe Airdroid, can help as well.