Hi chris orso,
So you should be able to connect the Time Capsule (via direct ethernet cable) to your new Mac mini and using the Migration Assistant, import files. I will though say, from personal experience, it can be a pain, as the Migration Assistant hasn't (in my own experience) been wonderful in fully restoring the User directory settings and User account. Instead, it oft' complains that the account exists and so, I have had to go through hours of restoring files via TM. This happens even if I have attempted the Migration Assistant restore to a fresh drive with new OS install and no User accounts. That said, take my experience with a grain of salt.
Thankfully you have the Time Capsule to restore from and so, if you haven't, read over Apple's page on Migration Assistant and Time Machine restore.
That said, good luck and post back if you've questions and the community will see if we can help.
Cheers,
Anthony