Can a person massage the lymphedema out of his arm?

Yes. The best treatment for this is manual lymphatic drainage and complete decongestive therapy. This is essentially a massage technique that stimulates the lymphatics and encourages the fluid to move towards the central venous system. Combined with compression this massage then controls the swelling. Please see a physician with special interest in lymph edema and seek this treatment.
There Is a machine. There is also a pneumatic machine that sequentially puts pressure that is pre set to drain the fluid out. It is more effective than the lymphatic massage. Our county health department has one for rent either for in home or in office use.
Yes. You can train in manuel lymphatic drainage and massage these areas start near clavicle, then armpits, upper arm, forearm, hands and fingers. Start at center of body work and finish at finger tips. This will retrain the lymphatics.