Renting an apartment in Yogyakarta this year

Try looking at the site rumah123

Google "Rumah disewa <area>" or
"Rumah dikontrakan <area>".

However, that won't always get you the best price as many places are never advertised anywhere except a little notice in the window of the property.
Here's an example from a free advertising site.

http://olx.co.id/iklan/disewakan-rumah- … ece9b1b3d4

That is far cheaper than your maximum, but be aware there is unlikely to be any security patrol or whatever. Much as Jogja is normally safe (and a very nice city), I wouldn't leave any expensive kit at home at night in such a place. That example is a little outside the city, about 30 minutes by bus, but there are plenty similar inside the city, commonly in the 'gangs', small side streets just big enough for a motorbike. The sense of community is usually very strong so crime is minimal.

This one is off the most famous street in Yogja, central to pretty much everything

http://olx.co.id/iklan/rumah-dikontraka … badedeb03e

OLX is always worth a look as many people prefer to advertise without using an agent, and you'll get a very good idea of real prices as it's aimed at locals, not foreigners.

The other option is spend a couple of weeks in an inexpensive hotel and get one of the staff to help you find a place.  You'll have to pay them a commission, but you'll probably end up with a far better price than trying to find the place yourself.

The commute is always the worst bit of a day, so get a map and see where your place of work is, then look for a place near there.

https://www.google.co.id/maps/place/Yog … 10.3694896

Yogja is a great place, crime is low and there's loads to do with your free time. Apart from the history, there are a few very nice beaches not far away, and the night life is sweet if you're into that sort of thing.

NOTE - Where ever you end up, make sure you find the local village head (Pak RT and RW) in order to get your residence documents.
You find much of what you need in the advice thread at the top of the Indonesia page.