It's definitely too dry, but it might not be gone. It may just lose it's leaves and then resprout when it's been made happy. You can check if it's still alive by taking a finger nail and gently scratching through the bark on the suspect branch or even the trunk. If you find green beneath, it's still good. Good luck!

I think it is snow rose - serissa. It could be a japonica. They like it on the dry side in winter but your plant does look too dry. Give it a good soak. If you are in a warm area take it outside in an open shade. It needs more light too. They do well in dappled Sun or part Sun.