There are many different types of dog shows: in the "Conformation Show," dogs compete based on how well their appearance conforms to a specific breed standard. These are the most common shows. In the "Obedience Trial Show," dogs compete based on how well they perform a specific set of tasks; in the "Dog Agility Show," dogs and their owners compete based on how they both handle an obstacle course. The "Field Trial Show" and "Tracking Show," are typically highly competitive events for hunting and tracking dogs. Finally, there are the offbeat shows, such as the "Novelty Shows" where dogs or both dogs and their owners compete in various categories, e.g., dog-owner look-alike contests, best-dressed dog contest, etc.
Except for the "Field Trial" and "Tracking" shows (because the Lhasa Apso is not a hunting or tracking dog), there's no reason why your Lhasa Apso cannot be highly competitive in any of these shows - competing and winning is just a combination of their breeding and training. In the most common of shows, the "Conformation" and "Specialty Breed" shows, judges evaluate dogs based on how well they conform to established, excepted, and published breed standards. These can be highly competitive shows at the "Championship" or "All Breed" level that focus on specific breeds. Interestingly, what these types of shows really do is promote and encourage the preservation of specific breeds across the generations. This really makes it a professional breeder's show rather than an average dog owner's show.
Your Lhasa Apso's standard originated in 1901 when Lionel Jacobs, a member of what was then known as the "Northern India Kennel Club," specified the characteristics of Lhasa as a distinct breed, separate from the currently known Tibetan breeds, i.e. the Shih Tzu, Pug, and Tibetan Terrier. This description has remained largely unchanged since then, and its purpose has been to guild breeders to preserve what is unique about this ancient Tibetan breed that has existed for thousands of years isolated in the Himalayas - basically, a watchful, intelligent, alert little dog with a highly distinctive coat, a guardian instinct, and a circumspect, tranquil, almost spiritual, demeanor. All of these characters made the Lhasa Apso perfect little watchdogs for the Dali Lamas and the Tibetan aristocracy.
The Lhasa Apso can also be highly competitive in the "Obedience Trial" and "Dog Agility" shows - it is just going to take a lot of work on the part of the Lhaso's owner and trainer. The obedience trials are a series of highly stylized and formalized maneuvers that the dog must execute as perfectly as possible. The relationship between the Lhasa and the trainer, in this case, is more of a dance partnership where both work to learn and execute maneuvers that are highly choreographed. The agility trials are a series of complicated obstacle courses, too complicated for any dog to manage on their own. Here the trainer must guide the dog by assessing the course, selecting strategies, and directing trade offs between precision and speed. This type of trial is even more of a partnership than the obedience trial.
In short, in the three most popular shows, the "Conformation, Obedience, and Agility Shows," there is no reason why the Lhasa Apso cannot be a formidable competitor against other breeds - indeed, their high intelligence probably gives it a natural advantage.