How much exercise does your Lhasa Apso need? Short answer: not much.
Longer answer: the Lhasa Apso is a cold adapted animal, which means it is great at conserving heat, and it does not have a lot of excess nervous energy, like other breeds, that it needs to burn off through exercise. However, the Lhasa Apso does need some exercise, and your Apso will enjoy spending time with you on 15-20 minute walks a few times each week.
However, there are some things to keep in mind: First, as is probably common sense, you should not walk or jog your Lhasa as you would a larger more long-legged breed. Additionally, breeds with short noses, like the Lhasa, can have trouble breathing when exercised vigorously - they are simply not built for it.
Second, just like humans, your vet should give your Lhasa a thorough check-up to assess its fitness level: heart, lungs, joints, ligaments, and weight. If your Lhasa has been sedentary for a long period, and if it is overweight, you will need to put your Lhasa on a diet and start out exercising slowly.
Third, you need to have trained your Lhasa to sit, to sit-stay, and come when you call. This is critical when walking your dog in heavy populated and trafficed areas.
Fourth, again as common sense, especially if your Lhasa is very young or old, start exercising gradually. Be consistent so your Lhasa can build up its stamina. Let yourself and your Lhasa warm up, to get your muscles lose and working, and allow for an equal "cool-down" period of time after you are done walking or running. Teach your Lhasa (or any dog) to follow your pace by walking quickly for 30 seconds, and then slow down for a minute, then speed up again. Mix it up - speed up and slow down for random intervals and at random speeds. Note this only works if your Lhasa first knows how to walk at your side and heel.
Some environmental considerations: Generally avoid rough terrain. Though the Lhasa is adapted for climbing and jumping, you'll still want to avoid possible trip and slip hazards. Gravel and rock can be quite hard on dogs' feet. After each walk, or especially if your Lhasa shows any change in their gait, check for any damages to the pads of its feet. Avoid glass fragments and other sharp debris during your outings.
Finally, some safety tips: Be aware of where your dog is at all times, especially in heavy populated and trafficed areas. Always use a leash - ALWAYS - unless your dog is exceptionally well trained, like a hunting dog. Firmly hold the leash - do not tie it to yourself. Finally, the leash can break and accidents can happen. Be prepared by making sure your Lhasa is wearing proper identification and that you have vaccinated your Lhasa for rabies.