Congrats on registering for a Warrior Dash! I've done the Mudathlon (same as the Warrior Dash, just a different company and charity) and it's so much fun! The good thing about the 5k-length races like the Warrior Dash is that the running is broken up so much that you will rarely be running for more than a quarter mile at a time. I didn't train for my race at all and even with crappy upper body strength, I survived to tell the tale!
Anyway, how is your current fitness level? How far can you run? (For Warrior Dash, you only really need to be able to run a mile or two without feeling like death afterwards. No biggie.) Can you do monkey bars? Can you pull your own body weight over a wall? Also, do you have teammates or will you be going at it alone? I did mine alone and was scared I'd get stuck at a high wall or something, but people are extremely friendly and helped me out. Another team even "adopted" me and ran with me through the mud across the finish line!
Bottom line: Just start small. Do some sprints, run a mile or two a couple of times a week, and work on your upper body strength if you are lacking there. You only really need to do body weight exercises because, well, you're only going to be lifting your body on the obstacles!
I'm doing the Tough Mudder (10-to-12 miles with obstacles just like the Warrior Dash) in May so I, too, will be spending the next few months training... I know what you're going through!