Step 1: Use the tarsal tunnel mnemonic.
The region behind the medial malleolus and in front of the Achilles tendon is the tarsal tunnel. Its contents, listed front-to-back, follow the mnemonic Tom, Dick And Very Nervous Harry: Tibialis posterior tendon, flexor Digitorum longus tendon, posterior tibial Artery, posterior tibial Vein, tibial Nerve, flexor Hallucis longus tendon.
Step 2: Pick out the artery.
Among those structures, the only artery is the posterior tibial artery (with its accompanying veins). It descends behind the medial malleolus and then divides into the medial and lateral plantar arteries to supply the sole.
Step 3: Exclude the other vessels by their territory.
$\bullet$ Anterior tibial artery runs on the front of the leg and becomes dorsalis pedis on the dorsum of the foot $-$ not behind the medial malleolus.
$\bullet$ Dorsalis pedis is palpated on top of the foot, lateral to the extensor hallucis longus tendon $-$ wrong location.
$\bullet$ Peroneal (fibular) artery lies on the lateral/posterior side near the lateral malleolus $-$ wrong side.
Step 4: Clinical anchor.
This is precisely the spot where the posterior tibial pulse is felt clinically, confirming the vessel that occupies this interval.
Step 5: Conclusion.
The artery passing behind the medial malleolus and in front of the Achilles tendon is the posterior tibial artery.
Final answer: Option 1 - Posterior tibial artery.