Hi Shnif!
Your plant is called Dracaena sanderiana, often called lucky bamboo, but is not a bamboo.
The roots wants a bit more air, but it can grow in water like you did. To clean up the upper part, you can cut off the dried out leaves. I think it is accumulated salts and maybe dry air making the leaves brown, and during winter, most houseplants really slow down, so they are not at their best. You can also totally remove that growing offshoot with a clean sharp cut, and the plant will just grow a new one, but got to have patience, it takes awhile. Wait for Spring to do that though. We are approaching Spring, so the plant will start perking up.
You can also clean up the container, remove carefully the rocks and plant, wash off and clean the rocks with water, so if there is any salt accumulation there it gets cleaned out. And replace it back in your container.
If you like, you can already move it to a container with drain holes and with well draining soil. It has a good root system already. Or in my case, since soil at times attracts fungus gnats, I maintained my similar plant in a glass container but I used water gel beads so the roots can still get moisture and allows good air circulation below. I don't use fertilizers it absolutely hates it.
