Based on the testimonies of Marina and Ruth Paine, it's assumed that LHO didn't have a Drivers License but Mrs. Paine was teaching him how to Drive. So it's probable that he knew how to drive by the Fall of 1963.
Several people claimed to have witnessed Oswald driving, including Albert Bogard who told investigators he inquired about buying a car 2 weeks before Kennedy's assassination.
Mr. BOGARD. A gentleman walked in the door and walked up and introduced himself to me, and tells me he wants to look at a car. I show him a car on the showroom floor, and take him for a ride out Stemmons Expressway and back, and he was driving at 60 to 70 miles an hour and came back to the showroom. And I made some figures, and he told me that he wasn't ready to buy, that he would be in a couple or 3 weeks, that he had some money coming in. And when he finally started to leave I got his name and wrote it on the back of one of my business cards, and never heard from the man any more. And the day that the President was shot, when I heard that--they had the radio on in the showroom, and when I heard the name, that he had shot a policeman over in Oak Cliff, I pulled out some business cards that I had wrote his name on the back on, and said, "He won't be a prospect any more because he is going to jail," and ripped the card up...
--------------------------
Mr. BALL. Did he appear to know how to drive the car?
Mr. BOGARD. Well, he had drove before, I'm sure, because he took off.
Mr. BALL. Did he----
Mr. BOGARD. He might have drove a little reckless, but other than that, he knew how to drive.
Mr. BALL. What do you mean, "He might have drove it a little reckless"?
Mr. BOGARD. Well, going 60 and 70 miles an hour right up a Freeway and took curves kind of fast.
Mr. BALL Did it appear to you that he knew how to handle the car?
Mr. BOGARD. Yes.
I'm inclined to believe the Bogard story is true.
Does it matter that Oswald claimed to have "some money" coming in a few weeks? If Oswald was part of some sort of conspiracy and expected to be paid a large sum of money, he might've considered buying a car shortly after. On the otherhand, maybe he was simply saving up to make a down payment on a car.