For as long as the release is supported you can use syspatch(8). They also appear to provide serial console access on request so you can perform a release upgrade via the recommended bsd.rd (RAMDISK) kernel method.
You can do unattended upgrades using autoinstall(8). There is a handy script for patching bsd.rd with a configuration file for the update process: https://bitbucket.org/semarie/upobsd/src although there should not be anything to configure on most boxes (maybe you want to deselect the X packages for your router).
There's also the manual upgrade method, for an example of what that looks like, here's the guide for going from 6.2->6.3: https://www.openbsd.org/faq/upgrade63.html#NoInstKern