Here is my full walkthrough of Act 4, complete with this achievement!
1. You're second stop on the boat will be the Rum Colony. Elect to go ashore.
2. Walk straight up to the bar, and stop at the bar. DON'T talk to Patch.
3. After like 15-20 seconds, he will answer the phone.
4. Select the option that hangs up, unlocking the achievement.