There are variety of choices once arriving Yogyakarta. Assuming you are Malaysian, hence able to speak Malay (similar to Bahasa Indonesia), here are the list :
(Arranged by budget)
(the most economical. safety might be an issue, especially in the two Yogyakarta's bus station or the train station. this is true even for locals who are not accustomed with the situation )
Key: Find buses that goes from Yogyakarta to Magelang. Borobudur is located in between. You may want to stop at "Mungkid" or "Muntilan". If you are from Yogya, you will get to Mungkid first. Then take "angkot", angkutan perkotaan (I can't find similar word in Malay), to the front of Borobudur. The same is from Muntilan. Beware that Borobudur park is large, so expect to walk about 15 minutes from the last stop. If you're lucky, you may be taken closer to the gate by the driver ...
Biggest and unsafest bus station is "Giwangan terminal". Pronounce in local: ngGiwangan Terminal. Every hour, there are big A/C buses to Magelang. You may need to sit close to driver, so easier for you to say stop at Mungkid.
Other bus station is "Jombor terminal". Though smaller and looks dirt you could expect to feel safer here. I think, there is direct bus to Borobudur, but don't know the schedule. The bus is small, we always refer to its size as 3/4 (tiga per empat). Remember, its only to mention the size, not to "bus company" which is printed on the body . Usually, buses go to Borobudur is not air conditioned. If there isn't available, you could always take Magelang.
Cost for above should be less than Rp 50,000, return.
It is more convenient to order taxi from Taxi booth in Airport. As the price is fixed and driver is registered, you may feel safer and not grumbling because of taking too expensive ride . If you stop it on the road, make sure you know the price first. My guess, it will be around Rp 150,000 to magelang, one way.
ALmost all Yogya taxi's, though all are metered, the driver wants to propose a fixed rate, which fortunately you could bargain.
The safest options. ANd I think the price is also reasonable. You may also combined it with Prambanan Temple, the biggest Hindu Temple.
Find information on tour operator in the airport or train station.
BTW. The closest temple from Yogyakarta is Prambanan. You could just take a "kol", which is a Mitsubishi L-300 type, from Bundaran Kampus (round about in front of Gajah Mada University boulevard, very famous spot).
Drive to Prambanan: 1/2 hours
To Borobudur: about 1.5 hours.