My basic suggestion for the best way to care for your Bowed Psaltery (or really any musical instrument for that matter) is to keep it in an environment where you would be most comfortable. In other words ... not too hot, not too cold, not to dry ... not too wet (or humid). Never leave your Psaltery in a closed up car on a summer day. Try not to expose it to direct sunlight for any length of time. Never take a shower with it (Ok ... just making sure you're paying attention!). I hope you're getting the idea.
You can store it in a case ... or its box but I like the idea of "displaying" it either in some type of stand (like a plate holder) or a wall hanger (we make them). One advantage is to make your Psaltery very easy to get to because you might be tempted to pick it up and play it on your way to doing something else ... which brings it right into the middle of your day (or evening) and, thus, your life. If it gets stored away in a closet or bookshelf somewhere you'll never get the same amount of enjoyment out of it. You might find it to be an interesting topic for discussion when friends stop by because it isn't an instrument most folks are familiar with. So ... learn at least one simple tune like "Morning Has Broken" or "What Child is This" that you can play for the curious ones.
I think you'll be surprised at how well the Bowed Psaltery holds its tune. It does tend to stay in tune even better if you play it more often! If you store the Psaltery "out" ... it might accumulate a little dust. The easiest way to dust it is to use a very soft bristled paint brush ... brushing parallel with the strings from the bridge toward the pointed end of the instrument. The tension on our horse hair bows is adjustable. You can release the tension inbetween playing sessions, but it is not necessary. Horse hair is subject to being eaten by a microscopic bug but it is easy to replace.