Correct. But if the car is off, u should be able to put in gear regardless of any condition(except completely destroyed tranny). The problem could be the release arm itself. Do u get lots of pressure when u lift the arm yourself? Or does it just slide up? If it just slides, then your arm could be bad, or the bolt that holds the release fork to the arm could be loose or something.
Also, yes..if it is the synchros, then more than likely first gear is the only synchro that's damaged.