I didn't really want to play this entire level with two controllers, so I spent a few minutes trying this solo and figured out the best way to do it.
First of all, this achievement is on New Alexandria. Load up the mission on any difficulty.
Theoretically you most likely can go for this achievement after clearing the hospital, but I highly recommend waiting until you at least get the objective to head to Club Errara so you can get a nice checkpoint right near the location.
You must play the entire mission and destroy all three jammers before the switch shows up. In the hospital, behind the generator you must destroy, is a jetpack. You need this for the achievement!
The building you are looking for is due west of Club Errara, off in the distance. Around the right side of it will be a button on the ceiling about 80% of the way up the building. You must first hit this button, and then fly both a Banshee and a Falcon through a ring-like structure that will spawn both of these silly vehicles. Keep in mind that when you hit this button, the "call for evac" button breaks and does not work, so you cannot lose your Falcon. After numerous attempts to hop out, hit the button, and hop back in before my Falcon falls into oblivion, I thought outside of the box. Please see the video below, I'll go step by step.
1. Locate the building in question due west of Club Errara.
2. Fly your Falcon to the top of this building and park it nice and safe. Hope off the side and jetpack yourself to the button. While jetpacking, you will need to hold your use button at the same time so the switch flips.
3. Now comes a tricky part. You see me walking back and forth on the skinny ledge. What I am doing is waiting until I see a Banshee appear on my radar. You need to jetpack around the corner and hold your use button to hijack it. It comes down to a bit of luck here, but I managed it after a few tries.
4. Now fly off to the ring-like structure mentioned earlier. I can't describe it's location well with words, so check the video. The structure is due south of Club Errara and the switch building. After hovering in between for a while, your Banshee should turn into a Phantom (1/2). If it doesn't, try changing your altitude in between the structure a bit.
5. Pilot the Phantom back to the building you just came from, and head to the roof where you parked your Falcon. carefully hop out of the Phantom (it will disappear) and jetpack yourself so you don't take fall damage like me.
6. Now fly your Falcon over to the same ring-like structure, and repeat step 4. It will turn into a Pelican! If you did all this on the same checkpoint (or somehow got a checkpoint in between vehicles) the achievement should unlock.
Let me know if you have any questions and if this was helpful. If it was not, please leave a comment below addressing the situation before downvoting. I'm just trying to help :)
Edit 1: Added important info about when the switch spawns, as well as clarified location of building a bit better (thank you XBFALCON