Bows and arrows originated before humans colonized the Americas. They appeared in Africa before humans colonized Europe and most of Asia in fact - but perhaps not before humans colonized Australia, where they were never used by pre-Contact natives.
I thought I should point out that bows and arrows being invented prior to the submerging of the Berring Strait does not mean that bows and arrows were not developed independently. For instance many civilisations developed the wheel, fire and a method for telling time independently of one another. It's possible that some groups borrowed the technology from others , whilst different groups developed it independently.
Speaking only to the question of the timing of the origin of bow and arrow technology in the New World, please see this post.
It seems that the prevailing thought is that New World natives developed the bow and arrow technology independently sometime between three and four thousand years ago. References are in the link.