Should one be alarmed if he/she finds out that they are in depression and stress together?, and what measures should they take to cope with the same

Often together: Stressful situations often trigger depression, and depression often leaves one less resilient in handling stress. So working with both is usually best. See someone--a counselor, therapist, psychiatrist. Lifestyle changes (exercise, diet, meditation) can helpa great deal; deeper understanding of causes and coping, will help additionally; medications if needed, too.