Your proposed sentence is an appropriate way to describe placing on its side in the morning, though note that in this case, 'put up' is being used to mean 'put away,' i.e. store in a non-usable state/position between uses. This meaning of the phrase is relatively common, but somewhat idiomatic, as it does not (normally) refer to actually lifting the object higher up. For the evening, you can use the same syntax; "put it down" or "put down my bed". "Place it down" would be an unusual turn of phrase.
If you wanted to describe these actions another way,"stand up" might also be used, as the taller orientation supported on two legs could be described as standing Likewise, "lay (it) down" could be used for the evening, or "lay (it) out". If confusion is arising using the "put up" terminology, "stand up" is more explicit, and will probably be more immediately clear to people unfamiliar with the situation. If you were referring to your bed directly instead of referencing the previous sentence (as in, "I put my bed down") you would omit the parenthetical it's, which would also be natural and appropriate.