Howdy Andrej,
That is easy enough to do. Just a little bit of PHP and HTML. That particular button appended to the filter ‘tribe_events_after_footer’. Although, it is set to not appear if (is_single() || !has_events()). ZThe CSS class to make it look like that is .tribe-events-button. Based on your screen shot, I am not certain if you are trying to move the prev event link there or not. If so, you might want to create a theme override to hide it from month/nav.php and use the function tribe_events_the_previous_month_link() to call it. But, keep in mind you would probably want do the same thing with the Next Month link as well.
This probably is a bit confusing if you have not taken a look at the code yet. Checkout views/month/nav.php inside of the-events-calendar plugin folder. Also lib/tribe-ical.class.php shows you the specifics for adding the export link, it is in the function maybe_add_link(). These sort of things tend to make more sense, at least to me, when I see it in action.
Does that answer your questions? Can I expound or clarify anything? Please let me know! Cheers.
– Brook