Disclaimer:
I don't have a time capsule and there may (or may not) be differences in restoring from a tm backup drive vis-a-vis restoring from a networked drive (which is what a time capsule is, is that right?).
But I will assume that Apple's setup assistant will recognize a time capsule and do a migration/restore from it just as if it were a "standalone" drive.
So... I would suggest this (in the order presented):
1. When the new Mini arrives, take it out of the box and set it up on the table, but DO NOT PRESS the power-on button yet.
2. Connect the time capsule directly to the Mini using an ethernet cable.
3. NOW press the power-on button for the first time.
4. Begin clicking through the setup process. At the appropriate moment, setup assistant will ask if you wish to migrate from another drive, a backup, etc.
YES, you want to do this.
5. So.... "aim" setup assistant at the time capsule. Give it time to "digest" everything... be patient.
6. Setup assistant should present you with a list of stuff to migrate. I would just accept the default... in other words, "everything".
7. Now, let setup assistant "do its thing", migrating all your stuff over. It will take a little while, so be patient.
8. When done, you should see the login screen, as you did before.
So... log in and "look around".
Do things look OK?
