You may want to see your physician first before starting an exercise program for this problem. Without knowing your age, past medical history, or the state of your general health, it's impossible to recommend specific exercises without a diagnosis.
Several different things could cause your symptoms. Women with undiagnosed heart disease sometimes report similar symptoms to what you've described. With their arms overhead, the heart is unable to pump enough blood uphill to supply the arms with needed oxygen.
This problem occurs most often in postmenopausal women with a personal or family history of high blood pressure or heart disease. A medical exam is needed to know for sure.
A second, more common cause of your symptoms could be a condition called thoracic outlet syndrome (TOS). This refers to pressure on the nerves and blood vessels as they travel down from the neck to the arms. For a variety of reasons, they can get pinched or compressed. Patients with TOS report neck and/or arm pain and fatigue, numbness and tingling, and muscle weakness.
The doctor can perform special tests in his or her office to look for TOS. A physical therapist can then help you with the right exercises. Sometimes stretching tight bands of tissue is all that's needed.
In other cases, strengthening and stretching the muscles is helpful. The therapist will show you how to improve your posture, if that is contributing to the problem. In very rare cases, surgery may be needed.