The carbs on these cars are a royal pain in the ass to work with. There's a system of over 30-50 vacuum lines, much of which eventually route to the carb. You can 'de-vac' them, but it can be trouble getting the stock carb to run correctly. Part of the reason these guys were so efficient was that when in good shape, the over-kill control on the carb allowed it to operate very close to that of EFI, able to tune for conditions and such. They're most common problem, and why this one won't idle most likely, is vacuum leaks.
Carb rebuild kits are not uncommon, I found one for my '86 at o'rielly's.. But, the carb itself is complex, more so than most US carbs, including the infamous Quadrajet.
If I recall right, because 84 was the 1st year, a lot of parts are unique to it vs. the other year 1G CRXs. It makes parts hunting on a car that's already a bit of trouble to find parts for even harder.
www.RockAuto.com will be your best friend (And CHEAP!), but finding parts in an emergency at a store can be a pain.
Beyond that, the cars are GREAT. Very fun to drive, very simple to work on, outside of the carb system. Upgrade to the 88-91 era front disk brakes, and enjoy a great little DD.