Hi there, thanks for your patience and happy new year! This one is tough to diagnose because there are a few different conditions that could lead to stress and browning in a succulent. Without knowing its home conditions and history it's hard to say definitively what could be causing this. It seems promising though, that the wilting has only happened on the outer, older leaves. Your plant should be able to bounce back from whatever stress it's under.

I would say the usual culprits that bring out the reddening/brown edges include light exposure, temperature, watering, and soil/pot conditions. Below are a few tips on each of these factors that could help you diagnose/make adjustments to avoid further browning:

Light exposure. If you noticed your plant turning colors after the long string of cloudy days we've had, it could be that your plant isn't getting enough light. You could supplement its light even with a fluorescent light bulb, just make sure the source is placed about 12 inches above your plant. On the other hand, if your plant isn't acclimated to receiving a lot of bright light/sun, it could be experiencing sunburn from being exposed to suddenly intense light, especially if it's young. In which case you may want to move it to another location if the light source is natural, and/or gradually get it acclimated to receiving more light day by day. Generally placing it near a south or south-west facing window is best.

Temperature. Also pay attention to the temperature of where your plants are placed. Unless your succulent is cold-hardy, extreme temperatures, sometimes even being placed on a very cold windowsill for extended periods of time, could be a source of stress. Or being placed under or over a heating vent where leaves could get dried out.

Watering. Succulents usually do best with infrequent, deep waterings. You'll want to let the soil dry out in between waterings. Either over- or under-watering could contribute to browning.

Soil/pot conditions. The last thing to check is your soil and pot size. Succulents could brown from not getting enough nutrients or on the other hand, too much. Typically they'll only need to be fertilized when they're actively growing (usually spring or summer time). And also make sure your plants have enough room to spread roots and grow in their pots.

Additionally, here's a link for more in-depth information on taking care of cacti and succulents: https://extension.umn.edu/houseplants/cacti-and-succulents

Bit of a long answer, but hope that helps. Good luck!