Will a "5.8Ghz" wifi antenna work well with routers broadcasting 5.1GHz?
The answer is likely, no, it wouldn't be suitable.

According to the datasheet, the workable frequency range is:


  5725 MHz to 5850 MHz


This corresponds to Wi-Fi channels 149 through 165. These are legal in several regions including the US/Canada, Russia, Australia, and others. They are not legal for high power applications in EU countries.

In any case it is highly unlikely that the antenna will provide the desired performance at 5.1GHz given that you would be operating over 600MHz outside the rated range of the antenna. The impedance this far out will not be nicely matched to its rated 50Ω impedance and so when hooked up to your Wi-Fi router will cause nasty signal reflections that will harm performance.

The same company sells antennae that are specifically designed for the 5.1GHz to 5.8GHz range, including ones with similar gain ratings (13dBi in one case). You would be much better investing in something which is rated for what you need.