I know this is an ancient post, but I am now in Belgrade dealing with exactly this issue. You see, I'm waiting for my CZ work visa, and due to strict schengen laws now in effect I cannot stay in CZ more than 90 days in 180.
To answer your question about finding a flat in Beograd, some of my advice includes this:
1. Most Serbs speak some degree of English--certainly more than Czechs. This is to your advantage when hunting for a flat.
2. First stay in budget hostels for first week or two, or find a room ('sobe' in Serb). These can be as low as 10 or 15 Euros in Beograd. In these first two weeks, start asking around, and using apartment rentl websites as you would anywhere else. One such website I found quite useful is-
mojkvadrat.rs
Yes, its in Serbian. Deal with it! No just kidding....if you use Google Chrome it has a nifty feature: it will ask if you want the website translated into English. Amazing! It actually works. If you have an old PC or old Mac you cannot use this feature I think...you must have Intel Core Duo processor and later version of Chrome for it to work.
So, first stay at sobe, and start looking for longer term rental. In Beograd you will probably pay 300 Euros per month for a 50 square metre studio flat, furnished, give or take.Â
Good luck in your search for your new home in Beograd! :-)