Warning: This is not CEA footage
So you start with the plasma pistol and shotty. You can’t pick up or use any other weapons. You have to get past about 15 Elites and swarms of Jackals and Grunts. And there’s an enemy Wraith. Are you going to painfully make your way down or speedrun and hope you’ll get lucky? Either way is going to be difficult, am I right marines? With this strategy, it will only take 10 minutes to get the Banshee. And there won’t be any battles being skipped other than 1-2 patrol groups.
Yeah sure, you could do it the speedrunning way. But where’s the fun in that? Once you get in the Banshee, it’s over. You win. You can alienate hundreds of Covies with that thing. That’s just like The Covenant and The Silent Cartographer mission start. You want to send this achievement out with a bang, am I right marines? If you like fighting your way to the Banshee, then you have come to the right place soldier.
Section 1:
But first, you gotta deal with the Sentinels, an Elite and his grunts, and another Elite with 2 Jackals. The safest and easiest way is to take out some Grunts with 1-2 frags, let the Sentinels do some damage to the Covies, hit the Elite with an overcharged bolt then take out 2-4 Sentinels with an overcharged bolt, finish off the survivors and then take out the reinforcements. No health loss.
Next battle is pretty simple so I won’t go into that. But I will provide a video for good measure.
Video for first encounter
Video for second encounter
Better first encounter video
Section 2:
Video for section 2
Now it’s time to get down the pyramid. Skip the ammo cache. Why? Because that gives you enough time to take the left side down.
Why take this path? Because it’s pretty easy to sneak up on the 2 Elites and the Jackal. You can assassinate them. You can get a plasma pistol with better charge. You can get plasma grenades. And you can take out the Covies on the right side more easily. Observe. 
Nothing beats being stealthy and getting assassinations on Elites.
[If you delay (by jumping) the checkpoint you normally get after finishing the second encounter, you can get it at the ammo cache but you might lose the opportunity to take the left path as the 2 Grunts will be patrolling that side]
Go on this ledge. From there, go here and then you can slide down and assassinate the Jackal but it’s better to wait until he reaches his destination so you have more time to assassinate 1 of the Elites. Then get a cp.
Section 3:
Video for section 3
Take your time with the other Elite and just slide down. As for me, I deal with the Grunts, Jackal, and Elites on the right side (which is why the video is longer than it should be. This section is very easy). Anyways, I recommend dealing with them and then sliding down. It means more grenades and another spare plasma pistol.
Finally, the first big battle of this level. This is generally a pain in the -Yoink-. At least 6 Elites including a commander gold Elite and a bunch of Jackals and needler Grunts.
First thing you should do is take out the Goldie. He spawns behind the rock my crosshair is on. It’s best to get a plasma grenade stick on him when he spawns. If you’re lucky, he’ll run into one of the Elites and they’ll both go out with a bang. If you find that difficult, try using the pincer movement on him: throw a grenade to the right side of the rock before allowing him to spawn then throw a grenade to the left side of it. If he still survives, finish him off with plasma pistol bolts. If he escapes, revert.
Now all that remains is several Elites, Grunts, and Jackals. On the other side is the Wraith and 2 plasma pistol Grunts. Just watch the section 3 video. Plasma grenades and shotty are the answer for dealing with the Elite majors and Elites in general. Use rapid plasma pistol bolt action against Grunts and Elite minors (they have weaker shields and they strafe less). You can also use plasma pistol bolts on Elite majors, but you should only do that to make them seek cover, like in the video. I managed to complete this part within 5 minutes by exploiting their preference: they seek cover too much. 
[An easy way to deal with the Elites is to get them to go where the sleeping Hunters were. Then you can stick them from above. Like this]
When you have defeated all the foot soldiers, you only have a Wraith to deal with. Get his attention then make your way to the Banshee without him seeing you, like in the video. If you did it correctly, he will be firing at nothing.
Get in the Banshee.
Achievement unlocked
10G - I’ll Give You My Shotty When You Take It From My Cold, Dead Hands
Hope this helped. 
