You are wise to research the best manner in which to store the wooden cross relief in order to reduce stress on it and ensure its long-term preservation. There are several potential stressors which could affect the cross, whether in storage or hanging on display, including the physical effects of gravity, and environmental risks of humidity, temperature, light, dust, pests, and possible risks from other materials with which the cross is in contact or with which it is in proximity.
My recommendation would be to store the wood cross relief flat, rather than hanging it in a vertical position. A custom fitted inert foam bed with an unbleached muslin liner in an archival storage box would allow the wood cross relief to be fully supported and eliminate the adverse effects of gravity. The joint between the two arms of the cross is at particular risk for damage if unsupported, and the ceramic accents attached to the cross may be at increased risk of the adhesive failing over time when it is hanging. Additionally, an archival storage box would protect the cross from light, dust, and pests. It may be best to store the cross and the mount separately, as each may have specific preventive conservation needs.
Regardless of how you choose to store the wood cross, ensuring against fluctuating humidity will reduce the primary environmental stressor for wood objects. A consistent relative humidity of 50-55 percent is ideal, but since few institutions can maintain this, sustaining as consistent humidity and temperature as possible will greatly increase the longevity of the wood cross relief.