I'm on the 4th A-Spec challenge right now and I don't have any cars in my garage that will fit it. It's about small Japanese cars (Lightweight K Cup). I have 19,000 CR and so I went to look in the dealerships and used cars but then I forgot the cars that I needed... so I had to go all the way through those menus to find it and try to remember one of these variations of the cars which may or may not even be in the used car shop. Is there ANY other way to review requirements without having to go all the way back to the actual race to see? I keep forgetting them and I really don't want to have to resort to writing something all these damn models down on paper. There should be a requirements check, or a tracker of some kind. Or even a way to redirect you to the car shop right from the requirements screen or something. Does anything exist?

Nope, don't think there's a shortcut in that case. Should be I suppose, but there isn't. 
 
Guess you're gonna have to write them down if you're unable to memorise a few car names (you know, pick a few to remember, seeing as you're only going to be buying one). Or just think logically and look for a Japanese car that looks small and light, then hop back over to make sure it's on the list, then buy it and you're done! It's all a learning curve mate, don't feel bad!
 
Also, if you get all Silvers in the B licence (fairly easy), you'll win an eligible car for the race that'll be faster than all the other cars on the track.

At least if you check your garage when in the menu for the particular race series, it'll only show what cars are eligible to race. I think thats a nice touch.

 I'm having the same problem but i found something that might help. The only catch is its for android users. Its a app called Gran Turismo car  Database, its a full list of all the car's with full stats in every GT game. you can search and apply filters and even add them to a little garage.  
Hope it helps. 
 
 
http://android-apps.com/applications/entertainment/gran-turismo-car-database/