You could try to connect the old one to the new one with an ethernet cable.
Then use setup assistant to "migrate" data from the old one.
How to do it:
1. When new one comes, hook it up to your display. DON'T power it on yet
2. Connect the old one to the new one with ethernet (I don't think a display is needed for what we're going to do, but I could be wrong)
3. NOW press the power on button on the new one and begin setup
4. Setup assistant will ask if you wish to migrate data from another Mac
5. "Point the way" for setup assistant to the old one.
6. Select the things you want to migrate (should be everything).
7. Let setup assistant do its thing.
8. When done, your new desktop should look pretty much like your old one did (because setup assistant did its job).
Warning:
Some of the apps from the old one may be too old to run on the new one.
You may have to update them.