Don't update a production site directly in production.
You WILL have a production site that is broken and endless days of stress that is totally avoidable.
Do it right.
Get a dev site up. Upgrade there. Test every aspect of your site after the upgrade.
You WILL need to upgrade most 3rd party extensions.
You WILL need to re-integrate your custom templates.
Once you are done, you can deploy to production.
Upgrades are fairly complex. There is no magic GO button.
It is up to you how you setup the dev and test sites.
Using docker can make that easy, and not take a lot of your time.
For the upgrade itself, download the core magento package and drop that into the dev folder version of your site.
Clear cache. ( or better use n98-magerun to effect the upgrade scripts )
(Please, don't do it direct on production, or else we might just see you scream for help on a dead production site next)