Strength, Not Strain
Read Ana Forrest's response:
This is an excellent question, one that many students face. Strengthening the shoulders and arms is good preparation for the more intense poses, such as Handstand and Scorpion; but strong shoulders and arms will also keep you from injury in poses such as Downward Dog and jumping back into Chaturanga Dandasana (Four-Limbed Staff Pose).
I suggest you begin by strengthening the surrounding muscles (rhomboid, scapula, trapezius, pectoral) and upper arms. A good pose you can use to strengthen this area is one I teach regularly, called Shoulder Shrugs. From a standing position, inhale and lift the shoulders up to the ears; keep them high and take them straight back. On the exhale, squeeze the shoulder blades together and draw them down the spine. Inhale again, spreading the upper back; exhale, squeeze mid-shoulder blades together, and draw them down. Inhale once more, broadening the upper back, and then exhale, squeeze the tips of the shoulder blades together, and draw them down. I often teach this from a Warrior II standing position and so repeat Shoulder Shrugs on the other side when I change legs. Teaching the back muscles to open and support the whole back is one key to a strong upper body.
Downward Dog, if possible with elbows bent (if that's too hard, do it with knees down), is another great pose to strengthen the shoulders and arms. Spread the hands wide and keep the elbows in line with the shoulders. Bend the elbows so they're one to three inches from the floor.
Sun Salutations can also help strengthen the shoulders. Avoid jumping back into Chaturanga until you can smoothly float back and keep the elbows in and the shoulders away from the ears. Build up to Dolphin Pose and Downward Dog on the wall; from there, Handstand is not far away.