YOu will have to remove anything that's completely dead, that's easy - but where you have branches with growth at the ends, and dead stuff nearer the base, I'd wait another week or so to see if new growth occurs lower down those particular stems - if not, then I'd ruthlessly cut it right back to encourage new green growth all over.

I'd also give it a light dose of a low phosphate fertilizer to encourage more growth from the base.

Good point - especially if its in a pot, can't quite tell from the pic. Growmore granules would do, or Miracle Gro general purpose