Detailed invoice won't matter.

Correct, Costa Rica departure tax is not included in the price of any tickets for flights departing Costa Rica. You can pay for the departure tax at many major hotels for a fee, which means you can avoid departure tax the line at the airport. The cost is slightly higher but worth it IMO. I tend to do this the day before departure so I don't have to rush and/or worry about delays when checking out. However, it sounds like you may not be at a major hotel since there is no phone where you are. If that's the case then get to he airport a bit earlier than normal. It's $29 and must be paid before checking in for your flight.