Technically speaking, Rip Van Winkle--whose story is set in the 20 years preceding and following the American Revolution--predates any notion of the "American Dream" because the idea and label weren't coined until 1931 by freelance social history writer, James Truslow Adams. Writing in The Epic of America, Adams defines his newly coined concept of the American Dream as:
that dream of a land in which life should be better and richer and fuller for everyone, with opportunity for each according to ability or achievement. ... It is not a dream of motor cars and high wages merely, but a dream of social order in which each man and each woman shall be able to attain to the fullest stature of which they are innately capable, ... regardless of the fortuitous circumstances of birth or position. (Adams, Epic of America)
I think you can see from the date of coinage and description that Rip predates the idea of an American Dream. Having said this, though, it is possible to identify the seeds of the later concept in the setting and circumstances of Rip's life. Consider this portion of Adams' definition that speaks of a new social order of freedom for all:
a dream of social order in which each man and each woman shall be able to attain to the fullest stature of which they are innately capable, ....
When Rip awakens, the American Revolution has come and gone past and been implemented successfully. People at the metamorphosed village inn, under a portrait of "General Washington" instead of King George, are talking about the new government and of new opportunities:
a lean, bilious-looking fellow, with his pockets full of handbills, was haranguing vehemently about rights of citizens- elections- members of congress- liberty ...
As liberty and freedom developed a new society following the Revolution, the newly coined American people found themselves in the midst of an unfurling dream that centered around personal and political freedom, liberty, and Congressional representation. At the conclusion of Rip's story, his greatest dream had been realised and he was free of Dame Van Winkle:
he had got his neck out of the yoke of matrimony and could go in and out ...without dreading the tyranny of Dame Van Winkle. ... [with] joy at his deliverance.