The Mysterious Man is a travelling salesman that will sell a variety of goods to the player, including a Lucky Sack (that has a high chance to contain a random Relic Piece), some cool and unique items, as well as the famous Wedding Ring.
In order to trade with the Mysterious Man, you first need to find him. This is easier said than done, because he is only in Portia a few days per month. He will always show up on Saturday the 27th and depart at the night between Sunday 28th and Monday 1st (of next month). This means that you can only trade with him during Saturday 27th and Sunday 28th.
So, where to find the Mysterious Man during these two days? He does not just rest in a fixed location, and instead spends his time in Portia travelling around the city. Luckily for us, we are able to track him on our map, so he should not be too hard to come across!
Here's his full schedule if you want to find him very easily. This applies for both Saturday and Sunday.
7:00 Arrives at Portia Harbor.
15:00 Arrives at Peach Plaza.
17:00 Arrives at Central Plaza.
18:30 Wanders from Central Plaza to his store Close to the Hazardous Ruins gate.
20:30 Arrives at Mysterious Man store. At this point it's too late to trade with him, so get to him before this time!