After completing the entire Main Story, you will be able to begin what is known as a Premium Adventure. At some point after beginning the Premium Adventure, you will receive a text message from the Kamurocho Newsletter. This message will explain how Newsletter has received an eyewitness report of a UFO in the skies above, which is the first thing you'll need in order to see the UFO for yourself.
After receiving said alert, go to the Ebisu Pawn Shop found in Kamurocho. Since finishing the game and receiving this text message, you should now be able to purchase the UFO Finder from the pawn shop in return for 10,000 Yen. Once you have it, equip it in your inventory and head to the rooftop of the Baseball Place also found in Kamurocho. By using the UFO Finder, you should be able to spot the first UFO in the skies above. After you do, head to the city of Onomichi and use the stair-lift to take you to the top of the mountain. It is here you will be able to see the second UFO with the help of the UFO Finder.