I was in a car accident on dec 31, driver in front of me stop right on the expressway, said that the driver in front of him was backing up. Which made him come to a quick stop, which made me have to come to a quick stop. So I slide into the driver in front of me. Police ask both of us were we okay we were both okay, I did not have any insurance at that time. Now the driver said that he has injury do to the accident. How long can you wait to go to the hospital or claim you were injury. And can I sue him for causing the accident.

Injury does not have a "time" frame, but it is common for injury from a rear end accident to take up to a couple days to even show up.
So he can claim it anytime he wants, and you will have to try to prove in court it is not true, and that means hireing an attorney since you don't have insurance.

And no you are at fault, don't matter if he stopped, as a police officer I see this all the time it is YOUR responsibility to be driving at a speed and at a distance that you can stop, if you hit him in the rear, it is always your fault.

So get your check book out and pay for his car and see if you can settle with him. If you don't have the money to pay for his damages, what happens, his insurance company will most likely pay his medical bills and his car repair if he has full coverage, and then his insurance company will sue you.

And why were you driving without insurance?

If this was on a limited access expressway and you can prove that someone else was backing up in violation of the law, that person may have a share of the fault. Since backing up on a expressway is illegal. However, you shouldn't have been driving so close to the car in front of you that you did not have room enough to stop. Therefore, the bulk of the fault rests with you.

If you can't arrive at an affordable settlement with this guy, you are in deep trouble.