Any idea how / where I can get this done? I am presently residing in Olomouc, Czech Republic..

Any computer shop can change the board for you -- whether it's worth the cost is open to question.  The board itself will run the equivalent of about $200 US.  Depending on what the labor costs, you may be facing a bill of 50% or more of what a new system costs.

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It's likely you may want to consider replacing the system rather than repairing it.

It looks like your system has a bad video chip - it will need a system board replacement.

Yes, I guess you're right... My laptop's anyway more than 4 years old, and $200 for a new motherboard is like more than 1/3rd the what I paid for it..

But why a whole new motherboard? Can't I just replace the graphic card? The ATI chip?

It isn't a card - it's a chip soldered to the mainboard, so while it can be replaced, you need a shop that can do micro-solder work.  You're likely to find the cost about the same as what a board replacement will cost - the chip will be cheaper than the board, but the labor charge will be much more.

At 4+ years old, you're better off replacing than repairing.