In my opinion, a ring of blinking or invisibility works when you put it on. It stays on until it is stopped (invisible person attacks) or the ring is taken off.
I think the text about working as the spell blink means just that, the effect it gives you is like the spell blink. It doesn't mean it has the casting time, duration, area of effect, or range of blink, just that when you put it on, you blink as per the spell description.
There are plenty of other ways to interpret it, but no rules, errata, FAQ advice, or Sage advice that handle it so it's pretty much whatever sounds right to your DM.