Yogyakarta is one of the main destination during Eid al-Fitr (Idul Fitri). Yogyakarta is the main destination for the Javanese who came back to their hometown for Eid. As such, they usually will come down to Yogyakarta for a short trip.

On the Idul Fitri's Eve, Muslims (which is the majority in Yogyakarta) are going out, saying prayers in the mosque, and some will held a march around town. All in peaceful and grand celebration and most of others visitor and residents alike will watch from the streets.

Flights and Train
Flights in and out of Yogya during the season may be fully booked, and  make sure you get your plane tickets  in advance. Flights will apply surcharge for high season period.

Train tickets will be sold out 3 months prior, and train delays are quite often during that period. The train will also apply surcharge for high season period.

Accomodation and Transportation
All hotels during the season may be fully booked (non-muslims also choose this national holiday season to get away). So make sure you book the hotel in advance. And expect the cost of accommodation to be MUCH higher than usual. The price usually rise one week prior and one week after Idul Fitri. 

Finding a car & driver for transportation can be quite difficult, even if you can get one the price will be quite high. You can contact several car rentals company or travel agent and ask for quotation for your travel date.

Store and Shopping Center
On Idul Fitri day itself, most shops will be closed for the whole day (some bigger shops and shopping malls will open after mid-day). Muslims will attend Idul Fitri prayer and afterwards it's the time to visit relatives and wish each other Idul Fitri greetings. 

Traffic may be more crowded than usual. 
Central Java and Yogyakarta will technically in standstill, due to heavy traffic (around 5 days before and after the Eid day).
The traffic will build up around main areas such as Malioboro and its surroundings.

Tourism Attraction
Main attractions such as Malioboro, Borobudur and Prambanan will be full of people.