Answer:
The key to this riddle is realizing that you do not have to pull the emergency switch in the first room, the entrance, and so that room will not lock down. You can leave it, and then return to it and go out the other door. And indeed this is exactly what you need to do so you can hit all 16 rooms and end in the
exit room.
Once you know to leave and reenter the first room, there are eight possible solutions to travel through all the rooms and end at the
exit. Because the first room is clear, you can backtrack through it to get back on the right track and pull the emergency switch in every room, saving the world from agonizing death by ancient virus.