Geinoch :
as long as I don't need to change the motherboard?
Considering this do you keep the cpu too.
Just upgrade the gpu to something nice like a gtx 760 or better.The psu has 4 rails of 18A and 4xpcie connectors from what i gather.It should be enough to power such a card.
If you wanted to be sure could you use both seperate power cables to plug into the new gpu so the power comes from two different rails (i think that each pcie cable comes from one of the +12Vrails).
For maybe others,found the info about the psu here,
http://allegro.pl/fsp-750-80-apg-fortron-4x-pc-express-gwarancja-i3942155488.html
You could also for money like that buy a nice case or/and a ssd and clone the o.s. or reinstall it.
A new psu could be payed from it too,although this one should do looking at the specs.Although the amount of cables tells a different story,but it's oem so maybe they figured that nobody needs more than those present.
I would first try things with the one you already have.
The color of the pcie slot can be different,it can be black too,it just has to look like the one i linked in shape.
This "TBGM01G1-1.1-8EKS3H motherboard." i cannot find.
If you open "dxdiag" by entering it in the searchbar in start or via "execute"(?) and open that will it in the tab "system" give the "system model" which is your motherboard.Can you give that info here.
Since it's an oem pc i think will it probably not be able to overclock,but it's worth a try if your open for it.
For really a $1000 could you get a complete new system if you wanted,but it's not strictly necessary.