How to reach Maison Degas from Capodichino Airport by bus:
take the shuttle bus " Alibus" (which leaves every 20 minutes; the cost is 4 euros if you buy it on the bus). It will bring you to the port in via Marina; there you can take the "R1" bus and get down in via Monteoliveto. Once you get there, you can reach the hotel on foot, walking Calata Trinità maggiore. 
How to reach Maison Degas from the Central Station (Garibaldi) by public transportation:
Take the metro line 1 in Piazza Garibaldi and get out at "Dante "stop. Once in Piazza Dante, just a few minutes walking (asking for Piazza del Gesù) and you will be at the Hotel.