The easiest option is to hire a taxi or car with driver from Yogyakarta. For budget travelers, take the Trans Yogya bus to Terminal Jombor, then catch a direct bus to Borobudur (about IDR 20,000). The journey takes around an hour and you will be traveling with locals. From Borobudur bus terminal, the temple is within walking distance.
Many hotels and guesthouses in Yogya can arrange sunrise tours that include transport, early entry to the temple, and return by mid-morning.