As I repeatedly mention, the most famous historical figure who caught the bubonic plague (most likely Y. Pestis) and could survive as well was Emperor Justinian the Great of Eastern Roman Empire during the outbreak also named after him, the plague of Justinian (541/2-544).

Concerning the outbreak of the Black Death from 1346/1347 to 1353, AFAIK few secular rulers in Europe officially caught the bubonic plague and died of this disease. On the other hand, many churchmen (including de facto ruler of the state) as well as members of European ruling families also fall victim to the plague. It is worth mentioning that we have surprisingly little amount of the reliable contemporary accounts of the plague across Europe than generally assumed.

The followings are some examples of the deceased during the Black Death:

• Joan Plantagenet (d. 1348), daughter of King Edward III of England: She had been on way to Iberia to get married with Prince Peter of Castile. One of the her entourages as well as ambassadors, Robert Bouchier, also died of this plague just before her.
• King Alfonso of Castile in Iberian Peninsula who was to be a father-in-law of the deceased Joan, also caught and died of the same plague in March 1350 (Kelly 2006: 269). Kelly notes he was the only known example of the death by the plague of reigning monarchs in Europe.
• King Pedro of Aragon also in Iberian Peninsula lost his wife, a daughter, and a niece in the same year as Alfonso in Castile [1350].
• Two archbishops of Canterbury (exactly speaking, one was to be consecrated after the election) was died of the plague, and they worked as a chancellor, the principal official among the central government of medieval England.
• Vasilii Kalika (Basil Kalika), Archbishop of Novgorod as well as de facto also political leader of the city who had defended Greater Novgorod [the city state of Norgovod in NW Russia, with surrounding territories] against the Swedish Crusade led by Magnus Eriksson, struck by the disease and died in 1352.

Reference:

• Kelly, John. The Great Mortality. New York: Harper Perennia, 2006.

Before other responses come in, you might be interested in my short response from another subreddit, where I mention rulers and other high-ranking people who died due to various illnesses, including bubonic and pulmonary plague.