Alright, so I called the city and figured I'd post my findings so if others were trying to find out what they have to do, someone has already checked (as of October 2017).
So, to begin, the City of St Paul does NOT have landlord licenses. So if you were confused about that as well, that's why.
Next, if you've purchased a property that has a Fire Certificate of Occupancy already, you'll need to submit a "Change of Ownership" form found here: https://www.stpaul.gov/departments/safety-inspecti...
If you've purchased a property that was owner occupied or wasn't a rental before, you'll need to submit the "Provisional Fire Occupancy Application" also found on the link above. I was told that as soon as the city clears the payment, you can then legally collect rent. She said that the Provisional Fire Occupancy Application MUST be renewed yearly until you've completed your Fire Certificate of Occupancy inspection. At the time I called, she said those inspections should happen within 1 year now, but before, they had to be scheduled out 3-4 years (ouch!).
As for the Landlord 101 class, it looks like you'll need to complete that as well for your Fire Certificate of Occupancy: https://www.stpaul.gov/departments/safety-inspecti...
Depending on the grade you received during your inspection, your property could be inspected every 6 years, or potentially every year if it's really rough.
TL;DR- City of St Paul doesn't have a landlord license. Properties are registered, not the owners. Must complete Provisional Fire Certificate of Occupancy and have payment cleared before you can legally rent (can do immediately). Then complete Landlord 101 class to meet requirement for getting Fire Certificate of Occupancy.