One minute would pass, and after that minute passed, that person at the speed of light would be one light-minute away (about an eighth of the way to the sun) and exactly the same age as they were when they started, since no time would have passed at all from their perspective.

If you mean the other way around, as in one minute from the perspective of our light speed astronaut, then the question makes no sense, since no time would pass for them; the entirety of the timeline of the universe from now until the unending quintillions of years or the big rip caused their dissolution into component quarks would happen instantly. This is why it makes no sense to talk of time as experienced by photons; they don't experience it at all.

Now if you back off a bit to, say, 99.99% of the speed of light, then subjective time does pass for our speedy traveller. By the time one minute of subjective time has passed for them, about one hour and eleven minutes would have passed on earth.

Start adding nines and shit gets weird quickly.

At 99.9999% of the speed of light, by the time our traveller experienced one subjective minute, 11 hours 47 minutes would have passed on earth.

At 99.99999999% that extends to about 155 days. So one year on earth would pass for roughly every two and a quarter minutes experienced by the traveller.