Regarding transportation from the airport to the apartments you have two options: metro or taxi.
If you choose the METRO, you shall take the blue line (Line Nr 3) which will get you directly after a 42 minutes ride to Monastiraki Metro Station. That is your stop. After getting off follow the exits to Monastiraki Square. Once you get out of the underground station and having Acropolis and the Monastiraki Metro Station on your back, you will find Ifestou Street on your left hand. After 3 minutes walk (230m) you will find the Apartments at number 41 (at the end of the street) on your left. Keep your eyes open for a big sign “MERCATO”.
A metro leaves every 30 minutes from the airport. The first leaves at 06.30 and the last at 23.30 (no service in between). A one-way ticket for one person costs 10 €, but there are also offers for families, small groups, day-tickets etc. The following link will show you how to get from Monastiraki Metro Station to the apartments: https://goo.gl/maps/X84N5xDtWz72
When in metro always mind your personal belongings (no wallets, cell-phones and valuable in your pockets, there are pickpockets who take advantage of the crowded trains.
If you choose to take a TAXI, the ride costs 44€ from 05:00 to 24:00 and 60€ at night time (24:00-05:00). The fare from Athens airport to Athens city centre is fixed and the price includes absolutely everything (luggage charge, tolls fee, VAT, meter start, airport charge etc.) so you don't have to pay anything extra. Since the apartments are located in the middle of a pedestrian zone, the taxi will not be able to bring you at our door. The nearest it can get is at Ermou Str 109 and Agiou Filippou Str, 80m away from the apartments (1 min walk) in front of the indian restaurant Mirch. So give the taxi driver this address to drop you off: ERMOU 109. The following link will show you how to get from there to the apartments: https://goo.gl/maps/XG2B722JHH82

With either means you choose you will still have to walk for a couple of minutes,, the taxi will drop you off just a little bit closer.
Regarding transportation from Piraeus port to the apartments you have again two options: metro or taxi.
If you choose the METRO, after a short walk right across the port piers you will arrive at the Metro station of Piraeus. There you shall take the green line (Line Nr 1) which will get you directly after 20-25 minutes ride to Monastiraki Metro Station. The Metro ticket costs 1.40 euros per person. Then you follow the directions as described above. A metro leaves every 5-10 minutes from the port.
If you choose to take a TAXI, the ride costs 20-22€ from 05:00 to 24:00 and 25€ at night time (24:00-05:00). The fare from Piraeus port to Athens city center is fixed and the price includes absolutely everything (luggage charge, tolls fee, VAT, meter start, airport charge etc.) so you don't have to pay anything extra.