First, I’m amazed nobody has asked how big the cans are!
Average store-bought canned cat food ranges between 3oz per can, up to approximately 5.5oz per can. Obviously that’s a huge difference, so this isn’t a question of "how many cans", but rather of "how much volume".

However! There’s the QUALITY of food to consider as well, which is what really counts. High-quality canned cat food with real meat as the primary ingredient will require less volume because the cat is able to more efficiently process the moisture and nutrients contained therein without having to pass through filler and grains. This means they will start to feel satisfied sooner, and eat less per sitting.

High-quality canned food is invariably more expensive than low-grade foods with lots of filler, but if health and longevity are you primary concerns, you want cat food with REAL meat (not byproduct) as the 2nd or 3rd ingredient listed at the absolute least.
First ingredient if you can get it.

Now the important part: as a rule of thumb, it’s recommended that for every pound your cat weighs, you should allow 20 to 30 calories of wet food intake. Pay attention to each food’s nutrition information, and you’ll notice that high-quality food actually provides more calories per ounce (you may need to check nutritional details online, as not all manufacturers list the caloric content on their cans), so you cat will have to eat even less again to satisfy his body’ needs.

As an example, take a cat who weighs approximately 10 pounds. If you want to MAINTAIN that weight, feed a solid 25-30 calories per pound, for about 250-300 calories per day. If you want them to lose weight, 20 calories per pound will help, so try to keep it around 200 calories/day.
Bear in mind that kittens, however, are advised to eat approximately twice the amount of calories as compared to adult cats, since they need that extra nutrition and energy to grow.

Lastly - and I cannot stress this enough - contrary to popular consensus, a cat does NOT require dry food. AT ALL. Period.
It does nothing for their health, and can, in fact, cause more harm. A feline’s digestive system is very straightforward and very efficient at extracting nutrients and water from the meat of their prey. This is why high-quality wet food with meat as a primary ingredient is so vitally important…in nature, cats only eat a very, very, VERY small amount of grains, typically less than 1-2% of their daily intake, and they can get that from the minimal grain and meal content present in all canned foods (if you’re an exceptionally dedicated cat owner you can even buy small pads of live pet grass for them to chew, usually for less than 5 bucks at your local pet store).
Excessive grain intake forces the cat to eat more empty calories in order to get the same amount of nutrition (leading to obesity, which can open the door to ailments like feline diabetes). it also makes them drink much more water to compensate for the moisture they are not getting from fresh meats, and this excessive fluid intake can place strain on their urinary system and kidneys, leading to potential renal (kidney) failure.
In short, cats simply are not designed, digestively, to be omnivores.
They are carnivores. Meat-eaters. That is literally all they need, and any vet worth their salt will tell you so. You may have to get your boy’s teeth cleaned about once per year to keep plaque buildup at bay, but that is a recommended yearly practice for ANY pet owner, regardless. Everyone should pay attention to a pet’s oral health, and that’s no different whether they eat dry, wet, or a combination of the two.

Anyway, I know this was a long read but hopefully it all helps a little to point you in the right direction. Calculate how much volume of food your cats needs, portion it out over 2 or 3 servings per day (depending on how much he wants to eat in one sitting), and go for it.
Your cat will be happy you did.

(P.S. - The advice posted by "Dj F" about weaning him off dry food slowly is absolutely true. Start with about 50/50, drop to about 25/75 after a week or so, then go 100% full canned after the 2nd week. This will give your cat plenty of time to get used to the new diet, both psychologically and intestinally.)
References :
1. Caring for an elderly cat (now 22 years of age) for the past 17 years.

2. Regular veterinary visits for adopted strays as a rescue-cat foster parent, from 3-week-old kittens to 18-year-old adults.

3. Extensive consultation with veterinarians at Hawthore Animal Hospital (Edwardsville, IL) on the topic of feline health and maintenance.

Are you actually seeing a hairball come up with the puke? At 13yrs old I would take him to the vet for a checkup because the puking is new and get the vets opinion on what food and how much.
I have heard from our vet /vet techs that some of the "indoor " formulas cause urinary problems down the road.
References :