You WILL get a check for $1000. I know because I just got one. They send you a form at the end of the year, and you are responsible for noting it in your taxes.
Same with $2000. Got one of those in Jan. In fact, the rep at the Lotto office in San Diego said anything up to $4999 gets the full amount. Above that, they subtract it from your check.
Congrats on the win!
