The old Monstera has lived elsewhere for some time. Often, when a plant is moved, it experiences a kind of shock. Everything is new to the old plant: the air, the light, watering, etc. This stress can lead to problems. Also, remember that an older plant will lose some bottom leaves. That is normal. If the yellowing leaves are higher than the base of the plant you may actually have a problem.
Swiss cheese plants, as well as most houseplants, need to be repotted every two to three years. The soil fails or the roots may overcrowd. Try repotting in a slightly larger pot with well drained soil. Use equal parts potting soil, perlite, and bark chips (the kind used for ochids). This mix will help drainage. If you see that the roots grow in a tightly packed circle it has become root bound. Prune away 1/3 of the roots with a sterilized sharp pair of scissors or clippers. Water the newly set plant well.
When watering make sure that water does not pool up in the saucer. The usual cause of leaf yellowing is over watering. It does not sound like you are doing that but if water stands in the saucer, it can wick up into the soil and cause root rot. Make sure that your new pot has drainage holes. If there are no holes in the bottom, retained water can damage the roots.
When you buy potting soil, read the label. Many soils on the market have built in fertilizers. If so, you don't want to add fertilizer. Yellowing leaves can also be a sign of poor nutrients so the new soil/fertilizer can help in that regard.
Make sure that the plant is not subject to chilly draft. Keep it away from heat registers, radiators, or air vents.
It may take a month or two for the plant to establish and get settled in its new environment. If you like, you can clip off a piece of the plant just below a new air root and place it in water. Change water every 3 - 4 days. Soon, new roots will grow and you can plant it! So even if you lose the old plants, you can get a new one started.