When & Where to Eat
Most people in Vatican City are tourists there for the day or are workers, most of whom live in Italy and only work in Vatican City. For both of these groups, few, if any meals are actually eaten in Vatican City itself so there is little semblance of a regular dining schedule. Having said that, when and where people eat tends to be similar to that of Italy.
If you're dining in Vatican City the options are limited as there are only a couple places for the public to eat. In the Vatican Museums there is a small self-serve styled eatery along with a pizzeria and a cafe. There is also a cafe near the Sistine Chapel rightfully called "Sistina."
Breakfast (or colazione) in Italy is generally eaten in the home and consists of fairly simple foods like cereals, breads, jams, and coffee. Most Italians eat this small meal after waking up, which varies from about 7:00 am to a couple hours later depending on the individual and his or her schedule. Lunch (or pranzo) tend to be much larger and takes much more time to eat.
Lunch tends to be eaten in the early afternoon and can go one for a couple hours. Eating out for this meal is somewhat uncommon still as pre-prepared meals and fresh homemade foods tend to dominate. However, in a city like Rome, eating out is becoming more common over lunch as paninis and pizza are some of the preferred choices, but again in Vatican City options are limited. Most workers will bring their own lunch, go home for a lunch, or eat in the cafe or pizzeria in Vatican City, while many tourists tend to grab some food from the Vatican Museum take away restaurant or grab a slice of pizza, of course when people eat in Vatican City is more dependent on their sightseeing schedule.
Afternoons tend to be interrupted with a snack, called a merenda in Italy, but in Vatican City this snack is less common. When it is taken it tends to be small and consist of bread, coffee, or gelato, which is no different from that of Italy.
Dinner (or cena) in Italy tends to be small and social. This is the meal to dine out, but not the meal to splurge as salads, soups, and cold appetizers are the most common items to eat. Dinner is usually eaten at about 8:00 pm or later, but with a heavy lunch and afternoon snack, few people are hungry before this time. In Vatican City dinner is not a common meal as the museums close at 6:00 pm. Only the people living in Vatican City itself eat dinner in the city and most do so in their home.
Staple Foods
There are no true staples in Vatican City itself as there are few restaurants and a continuously changing population and culture. Despite this, many of the foods are identical to the foods of Italy so one could argue pasta or bread is a staple. As a visitor to the country there are few places to eat, one of which is a pizzeria, so maybe pizza could be considered a staple food?
For the people that actually live in Vatican City, most are ethnic Italians so the food and its staples are identical to that of Italy, but Vatican City is quite ethnically diverse as well so what the staple foods are from individual to individual varies.
Specialties
There are no true specialties in Vatican City since there are few places to eat. Again, pizza is about as close as it comes.
Dining Etiquette
Since all eateries in Vatican City are catered to tourists, all are very relaxed so dining etiquette rules are informal. Additionally, there are no formal sit down restaurants in Vatican City so many of the traditional etiquette rules don't apply. Despite the almost entire lack of dining etiquette rules, dress tends to be formal in Vatican City as the churches and chapels require long pants and long-sleeved shirts are also encouraged, which is about as formal as dining in these public places gets.
For others, dining in Vatican City means you have some serious connections or are a diplomat on official business. If this is the case meals tend to be extraordinarily formal as all international dining rules will be required. This means you must eat in the continental style (fork in the left hand, knife in the right), your hands must be visible at all times, but your elbows should remain off the table as excellent posture is demanded. In fact, those last two rules should be followed even if you're only eating at the take out cafe or pizzeria.
Drinks
Vatican City has very limited options when it comes to beverages due to their limited number of eateries. Coffee is definitely the drink of choice as there are a couple cafes in the small country. Soft drinks and water are also accessible at the eateries and cafes.
Although wine is popular in Italy, Vatican City isn't known for their wine, although wine is always served at Catholic masses. There are some rules and regulations on what wine can be served; the wine must be alcoholic and it must be pure grape wine, but water is always added before the wine is served.
Lastly, many of the foods served in the restaurants are washed in the tap water. However, the tap water is safe to drink in Vatican City.