What is the best way to update the theme without losing my settings on the Theme Options?

Update via FTP

Please note:
make a backup of the Theme inside your WordPress installation folder /wp-content/themes/themename/

To update the Theme, login to ThemeForest, head over to your downloads section and re-download the theme like you did when you bought it.

Extract the zip contents, look for the extracted theme folder, and after you have all the new files upload them using FTP to the /wp-content/themes/themename/ folder overwriting the old ones (this is why it's important to backup any changes you've made to the theme files).

SNEWS Changelog

MightyMag Changelog

PowerMag Changelog

If you didn't make any changes to the theme files, you are free to overwrite them with the new ones without the risk of losing theme settings, pages, posts, etc, and backwards compatibility is guaranteed.

Alternative Solution:

If you prefer an automatic solution, you can use the Envato Wordpress Toolkit plugin as well. Since this is a third party plugin though its usage isn't covered by regular theme support.

Useful links: