Airport Train (Budgeted but a bit of hassle for transfer)
Airport train to the city (Yogyakarta train station), and then you have to take another public transport to the Amartya Jogjakarta Hotel. The whole journey could take up to 40mins (train) and 15 mins (from Yogyakarta train station to hotel)
Price:
Train ticket to the city is IDR 20.000,- for Reguler train) and IDR 50.000,- for the Express train (nothing much different only reserved/numbered seat and without stopping at any station for Express train).
Online taxi from Yogyakarta train station: Approx IDR 35.000,-/car or if you are solo traveler with light baggage, you can take a online bike as cheap as approx. IDR 20.000,-/journey
PS: the train ticket often sold out when purchase on the spot, which need to reserve/secure your ticket a day before, but if your flight got delayed, the train ticket can not be refunded/rescheduled, which you have to purchase the new ticket if still available.