Is supplicating by reciting a surah or an ayah or a specific supplication at a specific time and for a specific period, upon the occurrence of a particular problem, a sound Sunnah?

There is no specific supplication (dua) for solving problems in the manner mentioned. However, piety (taqwa), remembrance of Allah (dhikr), and reading the Quran are among the causes for solving problems and for the expansion of one's chest. Through piety, a person is provided for and finds a way out. Through remembrance, hearts find tranquility. Supplication is one of the most beneficial means for relieving distress. However, designating a specific time or manner for supplication and reading has no basis in evidence.