Will the FX 4100 bottleneck with Radeon HD7770 Crossfire?

No it will be fine. Bulldozer CPU's only bottlenecked high-end video cards like the 6970 (at it's time), 570/580 (at the time before 600 and 7000 series were released) and would have trouble with the newer generation of high-end cards like the 7970 ot 670/680.
 
Just to bump this up with a question. I am currently using an FX-4100 with HD 6770. Is it advisable to upgrade my gpu to hd7790 or the new r7 260x? Would it bottle neck the cpu? Or else I am thinking of getting a FX-6000 or FX-8000 series. Any suggestions and comments?
 
What about the r7 260x? Will it bottleneck my cpu or it wont? Because the price is almost similar to each other. In my country its like a RM30 difference..