Its a laptop. Your options for a video card upgrade are fairly limited.
There's a laptop specific series of 'removable' video cards called MXM. Sadly they're only remotely standardised, and you can't put an arbitrary MXM card into an arbitrary system. I didn't have much luck working out if this is an MXM card.
However, if the motherboards are the same, in theory, you could put a MXM card from the old motherboard into the new one.
"but the repair shop told me that I need an external graphic cards (is that even true?)." - More likely than not it should be a soldered on part if its needed. Else, you need to get a MXM card from the very same model of laptop, ideally.
"He also said that laptop graphic cards are hard to find and mostly i wont find a compatible one."
Kind of true.
That said, for a definitive answer, you need a service manual. I'll leave that as an exercise to the reader, cause I do sympathise since that is often an exercise in frustration, to the point where I google for this before I buy a laptop, 'in case' I need to do some off warranty work on it.
That said, I'd ask them if they could then remove the GPU from the old system and install it in the new one.
Nvidia GT555M
is soldered onto your motherboard, so the only way to replace it, would be to remove it. The process of removing it is not something you can do as an end user, and the equipment to remove it the correct way is 6-7 thousand american dollars.