I know there's only a few days left of the challenge at the time of posting this, but there's a pretty easy way to get this done and I thought I'd share it with you. There's one mission that can be A-Ranked in about four minutes, making the total completion time for the challenge under an hour.
Host a match on the Coral Lake East map, selecting the Bloated Anomaly mission. Select ANY Snow Wing dragon. Keep the difficulty on casual. Get 1 or 2 people to join you either by invite or random search. You can select an ampoule to help you out, as well. I'd probably use one with increased attack power, but it wasn't necessary for me.
There are three parts to the mission. During the first part of the mission you just need to kill the "balloons" on the enemy dragon's back as fast as possible. You need to kill them all in less than a minute for S rank on that mission part (S will offset a B Rank in another mission segment). I wouldn't worry too much about the projectiles coming at you, just focus down the "balloons" as fast as possible.
Next, you need to weaken the dragon's body, tail, head, and wings. During this part I would make my way up towards the far front of the dragon. It is fairly easy to get at least A rank on this part.
The final mission segment is shooting the eyes of the dragon. Keep an eye on the minimap to see where the eyes are open and vulnerable and shoot them as fast as much as possible before they close. I found making long passes out in the far front of the dragon to be the best location. It gave me good visibility on all the weak points, even those on the back and belly of the dragon. You may want to use a vision-type attack for this segment if you're going for the long-range strategy. Depending on your group you'll probably get an A or B rank here. If you managed an S Rank in another mission segment you'll still get an overall rank of A with a B rank on this last segment.
For the record, I got 5 A-ranks in a row on my first try with this mission (my first 5 A-ranks were from a handful of other mission). I hope this helps!