Concept:
The jugular foramen is an opening in the posterior cranial fossa through which important cranial nerves and venous structures pass. The cranial nerves passing through it are IX, X and XI.
Step 1: Recall the contents of jugular foramen.
The jugular foramen transmits:
\[
\text{Glossopharyngeal nerve (IX), Vagus nerve (X), Accessory nerve (XI)}
\]
It also transmits the sigmoid sinus continuing as internal jugular vein.
Step 2: Identify the nerve that does not pass through it.
The hypoglossal nerve is cranial nerve XII. It passes through the hypoglossal canal, not through the jugular foramen.
Step 3: Choose the correct option.
Therefore, the structure that does not pass through the jugular foramen is:
\[
\text{Hypoglossal nerve}
\]