will the r7 260x work with pcie 2.0 16x interface

There was a Tom's benchmark article on this exact topic. Actual in game fps with all things being equal except for 2.0 vs 3.0 was pretty much unmeasurable until you got up to top end cards in sli/crossfire. And that difference was very small.

 
Well ya, but if it fits in the OP's budget, and will run off of his current psu, so he doesn't need to spend any more money.........

Not everyone can swap mobo/cpu/psu/gpu,but would still like a bit of a bump to make gaming possible.

Short answer is yes it will work.