You could try to check in online 24 hours before your flight. If you're able to do so, then you shouldn't have to pay the departure tax "manually". However, if you're unable to check in online, you may have to pay the tax. At which point, if you're at a hotel where you can pay, you could choose to do so before heading to the airport.

And I went to prefunds.aa.com and looked at my old ticket, and even there I didn't see a line item with the departure tax broken out....