EDIT 2: Please go down to JDuke705’s solution below and give him a thumbs up for his amazing trick to find Play With Other missions that you actually want to play(like rescue the survivors)
EDIT: Thanks to this wonderful site: https://www.fortnitedb.com
we can track our stats! According to this, when I disconnected from a game and rejoined it did not count towards this. But it did count the mist monsters, survivors, structures built, and zone explored.
Thank you vl3tH4I1tYv for the link!
You can also check your progress via quests under your challenge tab. Thank you Greenway Gaming for finding that!
1000 "play with others" missions. As stated by APOPHIS1989, these are started by using the yellow question mark on your map. You can do these in any area, even if you’re a much higher level. You can do this with a party of 4 as well(thanks Tenshi3).
I've been pretty good at averaging 20 minutes per mission. This includes the 10 minute(sometimes less if you're lucky) storm shield defenses, and the 30-40 minute missions that happen due to randoms refusing to do the mission. My favorite missions to get besides storm shield defenses are "Defend the Grid", "Rescue the survivors", "Destroy the Encampments", and "Build the Radar Grid". These are the best in my opinion because no matter what the randoms are doing, the mission will end when the timer is up! With a coordinated group, exploring the map fully, saving survivors, and doing the mission can easily be done in less than 20 minutes for any mission. Have one person build defenses while the other 3 explore. I recommend staying in Plankerton because once you hit power level 40, this place provides a perfect balance of difficulty with number of mist monsters to kill.
Let's do the math:
Keeping an average of 20 minutes per session means this achievement will take you a whopping 333 hours! In order to get the 1500 zones explored it'll take 500 hours(and it'll take even more than that since any storm shield defense missions you get, you can't explore the zone).
Keep in mind:
Remember to build 500,000 structures while doing these missions. So try and place things down as often as you can.
Hope this helps!