Can exercise help control my gout flare ups?

No. Gout is a genetic condition. A patient with gout has an abnormal accumulation of uric acid crystal in their joints. No amount of exercise will prevent the abnormal accumulation in the joints. The most significant preventive measure for avoidance of gout flare ups is diet. Stay away from foods high in uric acid like red meat.
Maybe. I have never read anywhere that exercise can control gout attacks but by improving the circulation and hydrating the body after working out i think this will help. The best way to avoid gout attacks is by avoiding the foods that cause the formation of uric acid and staying well hydrated. One of my patients uses celery seed extract and said this has prevented him from having any more attacks.