Only

This is not a problem which is very frequently encountered, of course! It would be phrased to avoid the inherent ambiguity.

The sentence does not say specifically whether John has other cars or not. Therefore, it can mean either of the two. If it were said to me, and I were not allowed to ask for further details, I would guess (and I emphasize, guess) that it meant the first of your two choices.

1. If he just sold his only car, then he now has no car, correct.
2. This means what you say it means, but an English speaker would be unlikely to say it! Even less likely is "John just sold his only only-one-month-old car"!!!

John just sold his car, which was only one month old. (Here we're not sure if it was his only car.)
John just sold one of his cars, which was only one month old. (Here we know it was not his only car.)
John just sold his only car, which was only one month old. (Here we know it was his only car.)