Once you get far enough in the story, Guillermo will start offering you to do quests for the pub. These may range anywhere from acquiring and returning certain items to the pub or even killing a certain amount of monsters in upcoming areas. You'll start out at Rank E and need to progress up to rank S for the sake of this trophy and for the requirement of one Fury (Dohn on New Game + if going down Goddess Route).
The quests of importance are directly marked with an [Important] sub-tag and, when viewing their information, it will say [Times Cleared ##/01]. This means that, unlike most quests, they only have to be completed a maximum of one time to be permanently removed from your list.
Rank D Promotion Quest: Supplying Medicine [Important]
Type: Fetch
Item/Requirement: 5x Mid-Tonics, 5x Energy Shards
Description: You can simply buy these from Tomoe's Shop and then turn them in, if you haven't already got them in your supplies from completing quests and opening treasures in dungeons.
Rank C Promotion Quest: The Collector 1 [Important]
Type: Fetch
Item/Requirement: 1x Wildfire Spirit Stone, 1x Blizzard Spirit Stone
Description: The first real challenge of the quests to rank up come from trying to get the materials for these. This will require you to perform Synthesis and using the Item Radar to find the hidden items.
Wildfire Spirit Stone: 1 Blaze (5 Fire Stones from Yatagan Lava Flows) + 1 Lightning (Zawaza Plains)
Blizzard Spirit Stone: 1 Gale (5 Wind Stones from Bui Valley) + 1 Glacial (5 Ice Stones from Katticus Ice Cave)
Rank B Promotion Quest: Robot Research [Important]
Type: Fetch
Item/Requirement: 5x Ultra Powered Circuit
Description: Slay Red SPAs from the Lusamundo Ice Cave. Simple kill and return
Rank A Promotion Quest: The Collector 2 [Important]
Type: Fetch
Item/Requirement: 1x Voltremor Spirit Stone, 1x Slush Spirit Stone, 1x Firestorm Spirit Stone
Description: Quick way to get this is to sneak through the DLC Dungeons known as Kidanar East and the Wine Factory Ruins. You can find the stones as hidden treasures or in actual treasure chests (one can be found from a chest in Sedipua Den). It's much faster than having to get all of the stones to actually make them yourself. But I'd suggest doing it that way.
Rank S Promotion Quest: Mountain Monsters [Important]
Type: Kill
Item/Requirement: 3x Imperial Black Dragons
Description: Use a [Change Enemies] Fury in World Shaping on Zeppelia Valley and find these large, hulking dragons within the area. You only have to find and kill three to get this quest completed. Once the game registers that you are now Rank S, it will give you the trophy.