Yes, you are allowed to open the doors for adoration, but it should NEVER be exposed if nobody is there. I would talk to your pastor about exposing the Blessed Sacrament. DO NOT expose it, if he hasn’t given you explicit permission to do so. A lay person should only expose it for adoration, in the absence of a priest or deacon, for the faithful. If there are no faithful in the Church for adoration then it should not be exposed. I’m not too sure about exposing it just for yourself. I remember once, way back when I was an altar server and would serve at our weekly parish novena, with exposition by the priest. There was a snowstorm once and I was the only one who came: to help the priest with the incense and incense the Blessed Sacrament when he blesses the people. When we realized nobody was coming, he said he cannot expose the Blessed Sacrament for just the two of us and that there would need to be at least one lay person in adoration with us. We ended up praying to the closed Tabernacle, no exposition, for a few minutes and then he sent me home. So I don’t think you can just expose it for yourself and adore it with nobody present. Praying to a closed tabernacle doesn’t make your prayers any less important; its the intention that counts.
Also, I misread your post. I thought the Eucharist was locked up in the Tabernacle and you were taking out the Host for the people to pray to. I’m not too sure about locking up the Monstrance, with a host, in some cabinet, that’s not the Tabernacle. The Eucharist must be protected and one shouldn’t simply be able to open the cabinet and take out the Eucharist, for one could also steal it, or worse yet, desecrate it.