If you have daily sales, I’d update any listing quantities that change at least daily. If not, I’d update quantities every time you ship a round of orders.
Amazon looks at many factors in the listing ranking algorithm. In-stock/out-of-stock ratios and stock-on-hand vs sales velocity (which determines the days-on-hand to 0 stock forecast) all come into play and affect your listings’ search rankings. But, as far as updating your inventory numbers, I’d refresh them at the end of each shipping day to keep it all accurate to your actual stock-on-hand. Hope this helps!