IIRC, you only get to the quest to go into NSH after doing a complete run of E.Fleet.
You can definitely solo E.Fleet. No one really uses party finder because people either solo it or randomly join and leave parties any number of times after they go in the instance.
Get to know the dragon pulses and how they work. Red pulses usually put you at or near a boss. After familiarizing yourself with the map, try to arrive at bosses whenever you see that they're in combat = that means there's a group of people fighting it and you have a change to get your 1% damage in.
Once you've gotten the quest to go into Nightshade Harbor (the 24man version), get your Bloodshade Harbor purple soul shield ASAP, it improves your survivability in e.fleet and nightshade and their 6man dungeon counterparts (Blackram Supply Chain and Bloodshade Harbor respectively) and farming these instances becomes much easier.