Congrats on purchasing your duplex! My husband and I are in a very similar position, we purchased a duplex with a tenant on a month to month lease in April. We dropped off a letter the day we closed introducing ourselves and going over a few items, clarification on their rent, the utilities they cover, making sure we are included on their rent insurance as an interested party, as well as the items in the apartment they own. Ours was a little tricky because the tenant was friends of the seller so there was no lease, no security deposit and they own the fridge and dishwasher. We have kept the tenant on a month to month to see how they were, luckily despite horror stories you hear about inherited tenants he has been great! My husband lived in the vacant unit and we have been trying to fix it up to rent. As we get closer to getting the unit fixed up that my husband stayed in, we are going to increase the rent in the inherited tenants unit and also put a yearly lease with all the official documents together.
Do you have Brandon and Heather Turner's book on Managing Rental Properties? It is a fantastic read and offers an entire zip file of real estate documents.