For a month? You'll be fine. I did a year without much trouble. As already noted, Stefansson did a year without any trouble. People at ZIOH have done it for longer. Don't worry about cutting out fruit - fruit doesn't contain anything you can't get elsewhere.
A note on vitamin C. Vitamin C is required in collagen synthesis as a means to convert an amino acid into another protein which you need to build collagen. No vitamin C, no new collagen, you get scurvy. Except when you eat all meat, you don't get scurvy. Best theory as to why I've seen is that meat, being meat, contains a great deal of the proteins necessary for building collagen, so you don't need vitamin C to do the conversion, so you don't get scurvy. Ah, the beauty of meat. You don't need to eat raw meat, just rare or medium.
Doing ZC for a lifetime is a question mark because no one has actually done that in modern times with medical supervision (the bear half-counts but no one wrote a case study about him to review). But for a month? Your biggest problem will be dealing with adapting to a very ketogenic diet in such a short timeframe. It took me a while (3 months) to get into the swing of VLC. The first month was weird, as your body is adjusting to a pretty novel situation.
Anyway, good luck on your ZC month and don't worry about anything. Just don't overcook your meat and you'll be fine.