This quick guide explains how to obtain Mithridates crown in the Act 1.
Guide to Get Mithridates Crown
You have to pick going after the supplies instead of the ship when you get the smuggler choice.
Then you get it once you defeat him.
Note: If you pick the ship he escapes.
The dialogue will mention giving you his crown.
Important note: You wont be able to wear Mithridates crown until you’re in act 2.