Welcomepickups works well but I will suggest another cheaper and much more interesting solution when you want to immerse yourself in the atmosphere of Athens.
That's how I do every time.
At the airport you take the metro to Syntagma as you did before.
Then depending on where in Koukaki your accommodation is you can either:
1/ Stay in the metro and take metro line 2 (direction Elliniko) then get off at the next station "Sygrou Fix" which is at the eastern end of Koukaki
2/ Exit the metro, go to the Syntagma square bus stop and
take a trolleybus line 1 or 15, both lines have the same route in Koukaki and their lines cross this district with bus stops about every 250 meters.
I'm all for using public transportation--it is more interesting than taking a taxi and requires one to have one's wits about oneself. One really pays attention to a lot of what is going around one, while taxi riding is so passive. You learn about the place you're visiting when you walk and take public transportation, despite the greater inconvenience.
That said, in Lisbon, I was constantly floored by the fact that the buses did not have an illumined panel that showed the stops as they were coming up (I don't think my smartphone did, either, for the most part), and the signs were too small to be legible. So I almost missed bus stops or actually did. It was difficult to find where the stops were unless one knew the intersection or the area. And the Metro was undergoing repairs at one station and wasn't that even to navigate (three floors for different lines!) though extensive.
