**Core Concept**
The patient's presentation is consistent with acute appendicitis, a common cause of abdominal pain and vomiting, often involving the right lower quadrant of the abdomen. The clinical diagnosis is typically made based on a combination of symptoms, physical examination, and laboratory findings.

**Why the Correct Answer is Right**
The management of acute appendicitis involves surgical intervention to prevent complications such as perforation, abscess formation, and peritonitis. The correct answer is appendectomy, which is the surgical removal of the inflamed appendix. This procedure can be performed via open surgery or laparoscopically, depending on the surgeon's preference and the patient's condition.

**Why Each Wrong Option is Incorrect**
**Option A:** Non-operative management with antibiotics is generally not recommended for acute appendicitis, as it may lead to increased risk of complications and recurrence.

**Option B:** Conservative management with bowel rest and fluids may be considered in selected cases, but it is not the most appropriate management in this case, given the presence of a tender lump in the right iliac fossa.

**Option C:** Exploratory laparotomy may be necessary in cases of suspected perforated appendicitis or other complications, but it is not the primary management for acute appendicitis without complications.

**Clinical Pearl / High-Yield Fact**
It's essential to remember that the diagnosis of appendicitis can be challenging, and a high index of suspicion is required, especially in young adults. A classic presentation includes pain migrating from the periumbilical region to the right lower quadrant, accompanied by nausea, vomiting, and tenderness in the right iliac fossa.

**Correct Answer: A. Appendectomy.**