Stay at Lokman Setia Kawan! Nearby to Maliboro road. very nice decorated and nice staff. Consider as budget hostel, but trust me, it's good. I been there last year september, the superior room is around 200k rupiah/night. Cheaper room available.

I have been there last year, spent 2 days only, i went during raya (october), so i not sure bout may/june, guess should be alrite.
-for the 3 places, if u really hurry, make it 1 go:
3AM : depart from jogja to Borobudur for sunrise
8am : Depart from Borobudur to Gunung Merapi,
10am : Arrive Merapi, can go Pas Bebatan for 1-2hr trekking to view point.
2pm : Go to Prambanan

If you have 3-4ppl, that could be easily done by tour. For solo, difficult lo. Cannot do all in 1-day, just split it.  You can take public buses to Prambanan, it's just located between town and airport.
There is a lot tour package around.

-For backpacker/hostel, i recommend the Lokman Setia Kawan lo, that area all backapacker lodges, forgot name of street oredi.
There are a few lorong along the Jalan Maliboro and Jalan Sastrowijayan.

-IF want to do more, go Daeng Plateau for hot spring and relax.

-I hardly see night life there le, very iman city, but may be there are some hidden there.