**Core Concept**
The premature baby's symptoms of bullous skin lesions and periostitis are indicative of a condition associated with congenital infection, specifically congenital syphilis. This condition is caused by the transmission of Treponema pallidum from the mother to the fetus during pregnancy.

**Why the Correct Answer is Right**
Congenital syphilis is a significant cause of morbidity in newborns, particularly those born prematurely. The bullous skin lesions are a result of the infection's effect on the skin's epidermal layer, leading to blister formation. Periostitis, an inflammation of the periosteum, a layer of connective tissue surrounding the bone, is another characteristic feature of congenital syphilis. This is due to the infection's impact on the developing bone and surrounding tissues.

**Why Each Wrong Option is Incorrect**
**Option A:** This option might refer to a diagnosis of sepsis or a bacterial infection, which could also present with bullous lesions and periostitis. However, the clinical context and the specific symptoms point towards congenital syphilis rather than a bacterial infection.

**Option B:** This option could be related to a diagnosis of osteogenesis imperfecta, a genetic disorder that affects bone development and can cause periostitis. However, the presence of bullous skin lesions is not a characteristic feature of osteogenesis imperfecta.

**Option C:** This option might refer to a diagnosis of congenital rubella, another congenital infection that can cause a range of symptoms in newborns. However, the specific combination of bullous skin lesions and periostitis is more characteristic of congenital syphilis.

**Option D:** This option could be related to a diagnosis of congenital varicella, a condition caused by the transmission of the varicella-zoster virus from the mother to the fetus during pregnancy. However, the specific symptoms described are not typical of congenital varicella.

**Clinical Pearl / High-Yield Fact**
In cases of congenital syphilis, it is essential to consider the mother's past medical history, including any history of syphilis or exposure to syphilis. A high index of suspicion and thorough investigation are crucial to diagnose congenital syphilis early, as this can significantly impact the baby's prognosis and treatment outcomes.

**Correct Answer:** C.