Once my eczema clears up, is there anything I can do or take to make sure it doesn’t come back?

Moisturize! The key to keeping eczema from flaring is good routine dry skin care with moisturizers. Daily use of ceramide containing moisturizer such as cerave can maintain clearance. If you feel the sensation of itch before the flare, then already start oral antihistamines such as Claritin or zyrtec, much like how you would treat or prevent nasal allergies.
Protect. Avoid things that make you break out, soaps & wetness. Wash your hands only when necessary. Wear gloves when needed. Wear clothes made of cotton. Bathe only with a small amount of mild unscented soap, such as dove. Keep the water temperature cool or warm, not hot. Use the medicine your doctor gave you. Use a plain moisturizer daily. Avoid scratching or rubbing the itchy area. Manage stress.