This is a really fascinating question. Firstly, you are certainly correct that horses were introduced with the arrival of Europeans to the Americas. Before the arrival of the horse, Plains Indians lived in semi-sedentary villages, hunted buffalo, gathered plants, and grew crops such as maize, squash and beans, especially in the Southern sections of the great plains.

While the introduction of the horse certainly did make it easier to hunt buffalo, and made a purely nomadic lifestyle viable, it was done without the horse as well. The Olson-Chubbuck Bison Kill Site and Jones-Miller Bison Kill Site date back as many as 10,000 years and between show evidence of the hunting and butchering of 500 buffalo in a methodical, organized way. The typical method for killing bison without the horse was herding the buffalo into an enclosed corral, known as a buffalo pound, or herding them off of a cliff, known as a buffalo jump. Buffalo are nearsighted, so it's possible to get close to them funnel them into a pound or a jump by running them between low stone walls, using fire, or by encircling them in small groups. In this way, buffalo were a source of meat for Plains Indians, as well as a source of tools, clothes and shelter.

Once the horse become widespread, buffalo hunting became viable as the sole method of feeding, clothing and sheltering oneself and one's family. In this way, the horse culture with which we are more familiar with today rose to prominence. Although the horse was first brought to the Americas by Hernan Cortes, the start of the horse's dominance on the Great Plains was in 1680, when the Pueblo expelled the Spanish from New Mexico, and captured a large number of horses the Spanish left behind. By the 1730s, the Comanche were fully nomadic, and had become the great power of the southern plains, and by the 1770s, the horse was widespread across the northern plains, and buffalo hunting became an absolutely critical part of the way many tribes fed themselves.

To pose a direct answer to your last question--the evidence that we have as to what these cultures were like before they had access to horses is twofold--firstly, archaeological, and secondly written descriptions from Europeans who encountered them before the widespread horse culture, in the time between European contact and the 1730s. The conquistador Francisco Vázquez de Coronado y Luján is the first European whose record we have of contact with the Plains Indian tribes, around 1540. Henry Kelsey, an Englishman, witnessed an Assiniboine hunt in Canada in the 1690s, before the horse had arrived that far North. We do not have written records, however, from Plains Indians of that era describing how they hunted buffalo. In terms of archaeological evidence, there are numerous kill sites and buffalo jumps that we know of, ranging widely in the dates used, but going back as long as 10,000 years ago.