Despite what it looks like, you don't need to hurry, because the dropship won't attack your team. Don't forget to pick up the dog tag from the operations room, if you haven't already, then follow the other marines. The first encounter with the Weyland-Yutani mercenaries will take place in the area with containers, pictured above.
Depending on preferences, you can fire from behind cover, looking for opportunities for perfect shots, or take a more aggressive approach. In the former, consider using grenades and going through the left container (the above screenshot) to flank your enemies.
Enter the new building and go down the stairs. When you find yourself in the garage, search the area on the left to find a Legendary Weapon: Vasquez's Smart Gun. Same as with the regular smart gun, this one won't be permanently added to your inventory as well. Nevertheless, consider taking in with you for the further battles, especially since the game won't allow you to take it to the next mission.
Approach the exit of the garage. From now on you'll be forced to fight off both new mercenaries and regular Xenomorphs. Take advantage of the fact that the Weyland-Yutani soldiers will often be busy shooting aliens, so either surprise attack them, or even wait until one side falls.
After exiting the garage, start walking towards the hill outlined in the distance. I don't recommend going too fast, though, because you could suddenly find yourself under heavy fire or surrounded by Xenos. Be extra careful especially around the above area; there's a large group of enemies to deal with there.
Follow the path towards the elevator, eliminating whoever's left on the way. Once you reach your destination, examine the area thoroughly to replenish your supplies. When you're ready, take the elevator. Push the lever to end the mission.