Q.My lemon cypress starts turning brown on one side. The soil is moderately wet. Is the plant dying? What should I do to save it?
The plant was in a good condition for the first two weeks when it was brought home.

Certified GKH Gardening Expert
From the photo I don't see any browning of foliage, only what may be a little wilting of the foliar tips. So I would say at this point it is not dying.
This is most likely due to a water issue, too much or too little. How did you determine "moderately wet"? Did you use a finger or a moisture meter probe? Was that on the surface or throughout the soil in the pot.
https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/testing-moisture-in-plants.htm
Another reason for the disorder could be changes that occured with the move, or especially if it was transplanted.
I can't see from the photo if the plant is indoors or outdoors.If outdoors and you have had temps below 0 degrees F, it could be frost damage. Indoors it could be too warm.
https://www.gardeningknowhow.com/ornamental/trees/cypress/lemon-cypress-care.htm