This issue is a variant of The Amazing Spider-Man (Marvel, 1999 series) #583 [Direct Edition]. There exist further variants.
This cover has five variants by the same creative team; All have Spidey hanging upside down in the background taking a picture; 1st Print shows Obama facing right against a blue background; 2nd Print shows Obama facing left against a yellow background; 3rd Print shows Obama facing left against a blue background decorated with an American flag; 4th Print shows Obama facing left against a white background decorated with an American flag; 5th Print shows Obama facing right with the Lincoln Memorial in the background.
This is the same art as the Barack Obama variant cover 1st print.
This story, with 4 additional pages appeared in the digital only release Presidents' Day Celebration #1 from 2009.