Also a heavy computer user and I have terrible joints that hurt constantly. My tips (they aren't that great, but I'm still stitching even if it's less than I would like...) are as follows.
Check your ergonomics. Make sure you are staying relaxed while you stitch (no death grips). Look into getting a frame or another setup that lets you stitch with both hands.
Take frequent breaks. Massage your hands and stretch your arms and shoulders, look around the room and move your head and neck. You can find stretches and hand exercises specifically for crafters. I think they help with general stiffness.
Heat or ice your hands when you are done depending on what makes you feel better. Sometimes I also keep a hot tea near me to hold while I'm taking a break. Compression gloves might help if you want to try them.
You might also need to mix up your crafts and find some alternatives to do when your joints don't want to cooperate.
Good luck!!
You are making good progress! What is the final size for this going to be?