Istanbul, covering a huge area, is an incredibly crowded and complicated town. Inner city transportation is one of the biggest problems. If you don't know your destination it's very easy to get lost. Don't be confused! Once you learn to deal with the town's “unwritten rules” you'll have great pleasure. Public transportation in istanbul varies greatly. Municipality busses, public busses, inner city and general ships, car ferries, the tunnel, metro and suburban train systems all vary in price and have either tokens or ticket fares to pay. It is essential that you obtain an "akbil", a smart ticket key.
There are several transport alternatives to the Marmara Egitim Koyu, where the new campusof maltepe university is located.
From Maltepe downtown you can take a shuttle bus (either yellow shuttle or white one - ring). They stop by medicine faculty building (tip merkezi). All foreign students can use those buses for free.
Shuttle bus schedule: http://www.maltepe.edu.tr/en/shuttle-schedule
When travelling from Kadikoy, you can reach Maltepe University easily by 19b which belongs to iett (public transportation of Istanbul). You should mind the sign (Marmara Egitim Koyune gider) on the bus to be sure it is the right one. For this bus you can use your akbil.
Iett 19b schedule: http://www.maltepe.edu.tr/ulasim/iett.asp
Another way (especially if you miss the last bus) is to take a cab. The price should not exceed 15 tl from downtown Maltepe.
For campus map : http://www.maltepe.edu.tr/en/content/communication
For the public transportation schedules visit:
Public busses - http://www.iett.gov.tr/en/index.php
State railway - http://www.tcdd.gov.tr/tcdding/