This is really a two-part question:
- Which muscle keeps the neck upright?
- How does one strengthen it?
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It is fairly simple to strengthen it assuming you put in a consistent effort. To strengthen it in the best way possible, you should strengthen both the neck directly, and also the traps which support it.
To work your traps, the best way would be weight training. The most fundamental exercise that you can do in the gym to strengthen your traps is the shrug. You can do this with a barbell, smith machine, shrug machine, kettlebells, or dumbbells. Just load up a weight that you will be able to do 8-15 reps with, and do 3-5 sets. Basically just hold hold the weight, maintaining a neutral spine, elbows almost straight, literally just shrug like you normally would, but as high as you can, and hold for a second or two until you feel the burn, lower all the way down and feel the strech and repeat. Make sure to progress in some way with every workout, whether it is more weight, more repetitions, more volume or duration. There are also alot of exercises that indirectly work the traps like deadlifts and side raises so just look up on Google how to work the traps and you'll be fine.
As for the neck itself, you could wrap an exercise band around your neck, move far away until you feel the tension and simply turn your head from looking front to looking to your side for high reps for each side. You could also use something called a neck harness that's found in many gyms by putting it around your head, looking down then looking up. There's lots of ways to do this and you can easily look up many workouts for the neck online until you find a few that you like and work for you.
Lie on your back against a flat surface with your head resting on that same surface (i.e. not on a pillow). Lift your head as far as possible and hold for 5 seconds; repeat about 10 times. Do this again with your head pointed to the left and then to the right. Also, it's a good idea to work out your abdominal muscles, as they provide a foundation for the spine, which ultimately supports your head. (I cannot provide citations for this, as it comes from my physical therapist. I have a pretty bad neck, and this is the only thing that keeps my neck more or less pliable.)
You can also use a 30-inch towel to move your head from one extreme to another. Hold one end with your hand against the chest and the other end holding the towel with it wrapped around the back of your neck and around the cheek. Using the hand not against the chest, force your head in the direction away from the cheek it's against.
This helps me.
Edit: Make sure your back remains flat against the surface upon which you are laying.