It's not really made clear in-game, but once you've been to the dungeon and get it set as available in your Duty Finder, you don't need to stick around. Pick a day that you've got a significant amount of time available, queue for a group in DF, and go on about other things. When a group forms, a pop-up will notify you and you can port into the dungeon from anywhere. (Note: although you can be anywhere, it won't allow you to join while you're busy with something else, so be sure that you can quickly break off from whatever you're filling that time with as soon as the DF notification pops.)

It's quicker (at least as DPS) if you can get a pre-arranged group together, but not required. For pre-arranged groups, are you in a Free Company? Many (most?) FCs are pretty good about helping out members when they need to get through certain dungeons to advance the story (or for any other reason, for that matter).

Also, that quest is the start of a group of three group dungeons in a row. As soon as you finish the Sastasha dungeon for It's probably pirates, the game will next send you to Gridania for another group dungeon, the Tam Tara Deepcroft, after which it's on to Ul'dah for the Copperbell Mines. You're either going to have to go through the group-finding process three times or get a persistent group that has time to run one after another.