I've had a cat for most of my life, and "Demon" is my current kitty and has been with me for the 10 years I lived aboard and now the 2 in the house. She loves the boat! Recently we had the house sprayed for spiders (I don't do spiders) and I evacuated her to the boat which she hadn't been on for a few months. She was thrilled to be back and immediately went to verify that the essentials were there:
Litter box? Check.
Food dish? Check.
Scratching post? Check.
Demon used to fall overboard at the dock once or twice a year and be back out in seconds. At anchor I don't let her roam around the boat unless I'm aboard. We keep a long handled fishing net for cat retrieval if she were go over at anchor. Underway she stays below unless it's flat calm. Occcccasionally she will come up while under sail but will always stay near the companionway out of the wind.
The one time I had a mouse come aboard by running up my dockline Demon had eaten it by morning. Not before the little bugger had sampled 10 different food items though, but in the end he was the sample dish....
Yes you do need to go to the vet and get him/her fixed and get shots. If you wanted you could skip some of the shots because Velcro will essentially be an indoor kitty and won't be fighting with other cats. On the other hand it's better to get all the shots for paperwork/visa purposes. Keep all vet records.
You will want him/her fixed. If it's a he, he'll be less likely to spray inside. If it's a she, well, having a cat in heat on a small boat is quite the experience. Ask me how I know....
I'm not a fan like Bob is of the natural litter. CLUMPING cat litter is the way. It takes the urine and turns it into easy to scoop chunks. Poop is already in chunks, so you scoop the box but rarely have to empty it. The non-clumping litters let the urine soak to the bottom and you have to empty and scrub the box periodically.
I've read about a cruiser that used aquarium pebbles instead of litter. They would scoop out the poop and then dump the pebbles into a mesh bag and wash it over the side. Not sure if it would work but I've been wanting to try it some day....
I would get a litter box now and put the cat in it periodically to train them to use it. They are born with an instinct/preference for sand/dirt/litter but you might still want to get him used to it. You can dig in it too to show him.
One last piece of advice. Get yourself some natural fiber rope, like manilla or hemp. Wrap your mast on the inside of your boat with the natural fiber rope. It'll look good, insulate your boat from the cold mast, and the cat will LOVE the scratching post/tree that he can climb. My mast rope is still going strong after being a full time scratching post for 10 years. I also made a hammock out of string and a piece of fleece and some safety pins and put the hammock up on the ceiling by a portlight next to the mast. The cat would race up the tree and jump into the hammock where she was high, warm, and could see out the window. Kitties love hammocks.
Enjoy your new crew!
The lump on the bowsprit is "Demon". Stunts like this are why we now lock her below when at anchor, that an a bald eagle that was eyeing her once...