Is there any equivalent to FreeBSD's
portupgrade -R in Arch Linux?
-R --upward-recursive Act on all those packages required by the given packages as well.
The Arch Linux package manager,
pacman automatically manages dependencies.
ports you mean those third-party packages provided by Arch users in the unsupported repo (AUR), then some of the AUR helpers have an option to update dependencies. Yaourt, for example, will do this.
However, as the yaourt page on the wiki makes clear, it is neither wise nor advisable to completely automate this recursively.
Given the number of packages in Arch's official repos, this is not really an issue. There will be very few packages from AUR that have extensive dependencies, so upgrading them individually is neither difficult nor onerous.