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Many of these will do this naturally. It does not appear to be suffering from overwatering. It may not have enough light, as these are full sun plants, and require full bright sun all day. These will usually adapt to this over time though.

I don't see anything unhealthy just yet, but this article will help you to keep this plant healhty: https://www.gardeningknowhow.com/houseplants/swiss-cheese-plant/proper-care-of-a-swiss-cheese-plant-swiss-cheese-plant-propagation.htm

I am not aware of, and the internet isn't helping to identify, a plant with the common name of "Swiss plant". Since you are growing it as a houseplant, I am assuming you are speaking of Swedish ivy, Plectranthus australis. As the Latin name indicates, it is from Australia and nearby areas, not Sweden. Boy, are common names confusing! It likes damp but not wet soil so you must have drainage holes in the bottom or the roots will rot. On the other hand, it doesn't like soil on the dry side like succulents and lavender do. Here is an article on Swedish ivy care: https://www.gardeningknowhow.com/houseplants/swedish-ivy/swedish-ivy-care.htm

Review the article below on care requirements; watering and light.

https://www.gardeningknowhow.com/houseplants/swiss-cheese-plant/proper-care-of-a-swiss-cheese-plant-swiss-cheese-plant-propagation.htm

Yes, it is time to start thinking about potting it into a light seedling soil. There are not enough nutrients in tap water to allow for growth, but you don't want to over-do it. A seedling soil mix is light, yet contains just enough nutrients to help initiate rooting.

Here are some articles that will help you to care for the plant: https://www.gardeningknowhow.com/houseplants/swiss-cheese-plant/repotting-cheese-plants.htm

https://www.gardeningknowhow.com/houseplants/swiss-cheese-plant/proper-care-of-a-swiss-cheese-plant-swiss-cheese-plant-propagation.htm

https://www.gardeningknowhow.com/houseplants/swiss-cheese-plant/moss-pole-plant-support.htm

This plant is part of the philodendron family. Here's a link that will answer your question about what's happening to the leaves: https://www.gardeningknowhow.com/houseplants/philodendron/brown-philodendron-leaves.htm

Here's another article about the care of a Swiss Cheese Plant, that should be helpful:
https://www.gardeningknowhow.com/houseplants/swiss-cheese-plant/proper-care-of-a-swiss-cheese-plant-swiss-cheese-plant-propagation.htm

Your plant is probably experiencing a bit of transplant shock. I would give it some time, and use some of the tips in this article to help it: https://www.gardeningknowhow.com/plant-problems/environmental/learn-how-to-avoid-and-repair-transplant-shock-in-plants.htm. If it remains droopy, check the room temperature - too much heat can cause this plant to droop. Here's another article for you:
https://www.gardeningknowhow.com/houseplants/swiss-cheese-plant/proper-care-of-a-swiss-cheese-plant-swiss-cheese-plant-propagation.htm