Brachiation (Hanging Grip / Movement) Training
If you want a good chance of succeeding on the hanging obstacles on a Spartan Race, you're going to have to train your grip and learn to swing like a primate!
While being able to do pull-ups is great, it's not really necessary if you have a strong grip and good technique. It's also less important how long you can statically hang than it is to be able to grip HARD and release over and over again.
Combined with that, it is important to learn how to properly move your body in space while hanging to control your momentum and smoothly load / unload your hands. This can help you to be much more confident in transitioning from one grip event to the next as you move through the obstacles.
Put it all together, and you have brachiation. Wikipedia defines brachiation as:
"Brachiation (from "brachium", Latin for "arm"), or arm swinging, is a form of arboreal locomotion in which primates swing from tree limb to tree limb using only their arms. During brachiation, the body is alternately supported under each forelimb."
Here are a couple of ideas that can help you better prepare for the demands of brachiation and crush those hanging obstacles:
Step 1 - Side-To-Side Stationary Swing
This is the most basic skill for brachiation, and will help you learn both body control and the un-grip / re-grip pattern. This is an excellent demo video from Ido Portal's YouTube channel. Done correctly, it requires very little arm strength, only grip and momentum.
Note that while this is demonstrated on a fixed bar, you can (and should if possible) also practice this on rings and other hanging implements to vary the grip and coordination demands.
Start small with this and go for about ten swings, then work your way up to about a minute (or two) continuous, which would be a good approximation of the time required on most of the hanging obstacles on the Spartan Race if you're moving at a very deliberate pace.
Step 2 - Switch Grip Practice
Again, there is an excellent demo video for this from Ido Portal's YouTube channel. This is the next step once you've gotten pretty comfortable and proficient with the Side-To-Side Stationary Swing.
This will help you become more confident on certain obstacles (especially the Multi-Rig) where the grip targets may be moving / rotating and require different hand placements one after the other.
Again, if possible, mix up the implements you're using to train your un-grip / re-grip - bars, rings, ropes, different sizes and shapes all could be encountered on some of the Spartan Race obstacles.
Step 3 - Stop / Start Practice
Once you're strong, confident and proficient with the first two drills, you can amp up the body control element by stopping and restarting your swing a few times during each set.
Being able to control your body's momentum AND get going again from a dead stop can be a lifesaver if you happen to miss a grab in the middle of an obstacle, or lose momentum due to some other issue (E.G. someone stopping dead in front of you, or swinging into your lane).
Try to come to a smooth, controlled stop as quickly as possible, then restart your swing. If you're training on rings, it's also a VERY good idea to practice stopping with both hands on the same ring, and restarting your swing from there to reach the other one.
Further Work
If there are monkey bars or a long row of rings at a playground near you, give them a go! They'll help you practice coordinating the movement and un-grip / re-grip into forward motion. Just remember to un-grip when that side is UN-loaded (usually on the backswing of a movement) and bring that hand through to the next grip target.
Dead hangs from a bar or rings can be some good supplemental work for the above, and will help condition all tissues and joints involved to the unusual demands of hanging. Good news is, there is some research supporting regular dead hangs (of up to a cumulative total of 7 minutes daily) being very beneficial for shoulder and back health, so they're kind of a win-win!
A more muscularly-demanding variation is to practice the brachiating movements with a 90-degree flexed arm. While this is NOT the most efficient way to move, it does give you a little more room for error and time to recover if you miss a grip target because you won’t already be at full reach / extension on each hold. It can also help absorb a bit of the “grip shock” of each hand placement by taking a little force into the elbow joint. That said, it’s not strictly necessary for success. If you do want to build up to it, practice some 90-degree flexed arm hangs first, and then try and incorporate the flexed arm into your Switch Grip practice (above).
Nothing really replaces actually hanging for training your hanging grip, but heavy deadlifts and loaded carries (E.G. Farmer's walks) can definitely help build up your tolerance to grip demands. If you can, work up to at least 50% of your bodyweight in one hand, or your full bodyweight across both hands.
Git'Er Done!
Whatever option you choose from the above, as with your hill training, try and get some of this work in every other day, so 3-4 times per week, in addition to whatever other training you're doing.