On a browser level, you can use Firefox for Android which brings its own TLS library and currently supports the latest and greatest version of TLS (TLS 1.3).
This however, will not fix the handshake problems with apps that use the built-in SSL/TLS library. It is unfortunate, but your phone is reaching its end of life when it comes to modern security protocols. The only way to fix this would be to upgrade to Android 5.0, which comes with support for TLS 1.2.
I did some research and it seems like Android 4.1+ does support TLS 1.2 in client side TLS sockets, but that is disabled by default and has to be manually enabled by the app developer.
EDIT: K-9 Mail does exactly that.