To get the latest version of Safari you have to install the latest version of macOS.
Your Mac can be upgraded to macOS Sierra.
Mac Hardware Requirements
For details about your Mac model, click the Apple icon at the top left of your screen, choose About This Mac, then choose More Info. These Mac models are compatible with macOS Sierra:
- MacBook (Late 2009 or newer)
- MacBook Pro (Mid 2010 or newer)
- MacBook Air (Late 2010 or newer)
- Mac mini (Mid 2010 or newer)
- iMac (Late 2009 or newer)
- Mac Pro (Mid 2010 or newer)
https://support.apple.com/kb/sp742?locale=en_US
But that wouldn't solve the problem.
Quote:
Use Software Update to update or upgrade macOS, including built-in apps like Safari
https://support.apple.com/en-us/HT201541