This page of the Weird West guide explains how to obtain a horse, used not only as a means of transport but also to store addiitonal items in the saddlebags.
Time in Weird West passes dynamically, which means the time of day changes whenever you play through quests or travel on the world map. All NPCs and shops are available during the day, but you won't be able to do much business during the night.
Horses in Weird West can be purchased at the stables for around 290 dollars. Some locations of Weird West's map are located very far apart, so the travel may take up to two days sometimes. Owning a horse reduces that time by more than a half.