Wordpress: seamless integration of a new theme to a working site

We have a WP news site created with a theme called Ionmag 2.1 that was installed about 2 years ago. We need to seamlessly apply a new theme called NewStar by Bold Themes, without breaking anything and losing any of the old functionality and stats (including social media stats and shares per page). In the process there must also be an optimization of the CMS, in order the site to run faster. The portal language is Greek. There is no problem if the site goes down for a few hours, even for a couple of days. The only thing that matters is the job to be done properly. After the new installation, the site must look exactly like in the New theme" pdf attached.