You can find bike/motorcycle/scooter rental in downtown or backpacker street. There are so many motorcycle rental. They offering matic or semi auto motorcycle.
The rate about IDR60 000 per day or Rp60 000/day.
If you rent for 1 week/month, you will get cheap price.
*) Don’t forget to ask brown paper (letter for scooter/motorcycle) / STNK.
**) Better you have international drive license. But, if you don’t have drive license, it’s ok since you drive in Yogyakarta province area. The police won’t catch you. Unlike Bali, the police here more friendly to the international traveler/tourist.
***) Check the letter, is tax paid for a year. Otherwise, police will catch you and ask some money. It will happened in the outside of Yogyakarta province such as Muntilan, Magelang, Purworejo, Klaten, Solo, etc.
****) The most important before you rent a scooter, make sure that you are get used to Asia (crazy) traffic. So, probability of accident can be reduced.
HERE THE MAP OF SCOOTER RENTAL POINTS WHICH IS NEAR TO THE AIRPORT, TRAIN STATION, ETC.
*) I recommend “Pamitran” scooter rental for traveler. I have checked the office and their service. It more than other scooter rental can be offer. Here some reason:
1. Free charges drop off the scooter to your hotel. Even you ask to the airport, they will do.
2. Friendly service operator and able to speak in english.
3. Scooter condition in good performance.
4. Helmet washed in every week.
5. Experience since 2009.